src/HOL/SEQ.thy
 author wenzelm Tue May 04 20:26:53 2010 +0200 (2010-05-04) changeset 36650 d65f07abfa7c parent 36647 edc381bf7200 child 36663 f75b13ed4898 permissions -rw-r--r--
fixed proof (cf. edc381bf7200);
```     1 (*  Title:      HOL/SEQ.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Jeremy Avigad
```
```     5     Author:     Brian Huffman
```
```     6
```
```     7 Convergence of sequences and series.
```
```     8 *)
```
```     9
```
```    10 header {* Sequences and Convergence *}
```
```    11
```
```    12 theory SEQ
```
```    13 imports Limits
```
```    14 begin
```
```    15
```
```    16 definition
```
```    17   Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
```
```    18     --{*Standard definition of sequence converging to zero*}
```
```    19   [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
```
```    20
```
```    21 definition
```
```    22   LIMSEQ :: "[nat \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
```
```    23     ("((_)/ ----> (_))" [60, 60] 60) where
```
```    24     --{*Standard definition of convergence of sequence*}
```
```    25   [code del]: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```    26
```
```    27 definition
```
```    28   lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
```
```    29     --{*Standard definition of limit using choice operator*}
```
```    30   "lim X = (THE L. X ----> L)"
```
```    31
```
```    32 definition
```
```    33   convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    34     --{*Standard definition of convergence*}
```
```    35   "convergent X = (\<exists>L. X ----> L)"
```
```    36
```
```    37 definition
```
```    38   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
```
```    39     --{*Standard definition for bounded sequence*}
```
```    40   [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
```
```    41
```
```    42 definition
```
```    43   monoseq :: "(nat=>real)=>bool" where
```
```    44     --{*Definition of monotonicity.
```
```    45         The use of disjunction here complicates proofs considerably.
```
```    46         One alternative is to add a Boolean argument to indicate the direction.
```
```    47         Another is to develop the notions of increasing and decreasing first.*}
```
```    48   [code del]: "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
```
```    49
```
```    50 definition
```
```    51   incseq :: "(nat=>real)=>bool" where
```
```    52     --{*Increasing sequence*}
```
```    53   [code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
```
```    54
```
```    55 definition
```
```    56   decseq :: "(nat=>real)=>bool" where
```
```    57     --{*Increasing sequence*}
```
```    58   [code del]: "decseq X = (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```    59
```
```    60 definition
```
```    61   subseq :: "(nat => nat) => bool" where
```
```    62     --{*Definition of subsequence*}
```
```    63   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
```
```    64
```
```    65 definition
```
```    66   Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    67     --{*Standard definition of the Cauchy condition*}
```
```    68   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```    69
```
```    70
```
```    71 subsection {* Bounded Sequences *}
```
```    72
```
```    73 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
```
```    74 unfolding Bseq_def
```
```    75 proof (intro exI conjI allI)
```
```    76   show "0 < max K 1" by simp
```
```    77 next
```
```    78   fix n::nat
```
```    79   have "norm (X n) \<le> K" by (rule K)
```
```    80   thus "norm (X n) \<le> max K 1" by simp
```
```    81 qed
```
```    82
```
```    83 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```    84 unfolding Bseq_def by auto
```
```    85
```
```    86 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
```
```    87 proof (rule BseqI')
```
```    88   let ?A = "norm ` X ` {..N}"
```
```    89   have 1: "finite ?A" by simp
```
```    90   fix n::nat
```
```    91   show "norm (X n) \<le> max K (Max ?A)"
```
```    92   proof (cases rule: linorder_le_cases)
```
```    93     assume "n \<ge> N"
```
```    94     hence "norm (X n) \<le> K" using K by simp
```
```    95     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```    96   next
```
```    97     assume "n \<le> N"
```
```    98     hence "norm (X n) \<in> ?A" by simp
```
```    99     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
```
```   100     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   101   qed
```
```   102 qed
```
```   103
```
```   104 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```   105 unfolding Bseq_def by auto
```
```   106
```
```   107 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```   108 apply (erule BseqE)
```
```   109 apply (rule_tac N="k" and K="K" in BseqI2')
```
```   110 apply clarify
```
```   111 apply (drule_tac x="n - k" in spec, simp)
```
```   112 done
```
```   113
```
```   114 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
```
```   115 unfolding Bfun_def eventually_sequentially
```
```   116 apply (rule iffI)
```
```   117 apply (simp add: Bseq_def)
```
```   118 apply (auto intro: BseqI2')
```
```   119 done
```
```   120
```
```   121
```
```   122 subsection {* Sequences That Converge to Zero *}
```
```   123
```
```   124 lemma ZseqI:
```
```   125   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
```
```   126 unfolding Zseq_def by simp
```
```   127
```
```   128 lemma ZseqD:
```
```   129   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
```
```   130 unfolding Zseq_def by simp
```
```   131
```
```   132 lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
```
```   133 unfolding Zseq_def Zfun_def eventually_sequentially ..
```
```   134
```
```   135 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
```
```   136 unfolding Zseq_def by simp
```
```   137
```
```   138 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
```
```   139 unfolding Zseq_def by force
```
```   140
```
```   141 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
```
```   142 unfolding Zseq_def by simp
```
```   143
```
```   144 lemma Zseq_imp_Zseq:
```
```   145   assumes X: "Zseq X"
```
```   146   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
```
```   147   shows "Zseq (\<lambda>n. Y n)"
```
```   148 using X Y Zfun_imp_Zfun [of X sequentially Y K]
```
```   149 unfolding Zseq_conv_Zfun by simp
```
```   150
```
```   151 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
```
```   152 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
```
```   153
```
```   154 lemma Zseq_add:
```
```   155   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"
```
```   156 unfolding Zseq_conv_Zfun by (rule Zfun_add)
```
```   157
```
```   158 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
```
```   159 unfolding Zseq_def by simp
```
```   160
```
```   161 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
```
```   162 by (simp only: diff_minus Zseq_add Zseq_minus)
```
```   163
```
```   164 lemma (in bounded_linear) Zseq:
```
```   165   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
```
```   166 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   167
```
```   168 lemma (in bounded_bilinear) Zseq:
```
```   169   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   170 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   171
```
```   172 lemma (in bounded_bilinear) Zseq_prod_Bseq:
```
```   173   "Zseq X \<Longrightarrow> Bseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   174 unfolding Zseq_conv_Zfun Bseq_conv_Bfun
```
```   175 by (rule Zfun_prod_Bfun)
```
```   176
```
```   177 lemma (in bounded_bilinear) Bseq_prod_Zseq:
```
```   178   "Bseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   179 unfolding Zseq_conv_Zfun Bseq_conv_Bfun
```
```   180 by (rule Bfun_prod_Zfun)
```
```   181
```
```   182 lemma (in bounded_bilinear) Zseq_left:
```
```   183   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
```
```   184 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
```
```   185
```
```   186 lemma (in bounded_bilinear) Zseq_right:
```
```   187   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
```
```   188 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
```
```   189
```
```   190 lemmas Zseq_mult = mult.Zseq
```
```   191 lemmas Zseq_mult_right = mult.Zseq_right
```
```   192 lemmas Zseq_mult_left = mult.Zseq_left
```
```   193
```
```   194
```
```   195 subsection {* Limits of Sequences *}
```
```   196
```
```   197 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
```
```   198   by simp
```
```   199
```
```   200 lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> (X ---> L) sequentially"
```
```   201 unfolding LIMSEQ_def tendsto_iff eventually_sequentially ..
```
```   202
```
```   203 lemma LIMSEQ_iff:
```
```   204   fixes L :: "'a::real_normed_vector"
```
```   205   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   206 unfolding LIMSEQ_def dist_norm ..
```
```   207
```
```   208 lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```   209   by (auto simp add: LIMSEQ_def) (metis Suc_leD zero_less_Suc)
```
```   210
```
```   211 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
```
```   212 by (simp only: LIMSEQ_iff Zseq_def)
```
```   213
```
```   214 lemma metric_LIMSEQ_I:
```
```   215   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
```
```   216 by (simp add: LIMSEQ_def)
```
```   217
```
```   218 lemma metric_LIMSEQ_D:
```
```   219   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```   220 by (simp add: LIMSEQ_def)
```
```   221
```
```   222 lemma LIMSEQ_I:
```
```   223   fixes L :: "'a::real_normed_vector"
```
```   224   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   225 by (simp add: LIMSEQ_iff)
```
```   226
```
```   227 lemma LIMSEQ_D:
```
```   228   fixes L :: "'a::real_normed_vector"
```
```   229   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   230 by (simp add: LIMSEQ_iff)
```
```   231
```
```   232 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
```
```   233 by (simp add: LIMSEQ_def)
```
```   234
```
```   235 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
```
```   236 apply (safe intro!: LIMSEQ_const)
```
```   237 apply (rule ccontr)
```
```   238 apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
```
```   239 apply (simp add: zero_less_dist_iff)
```
```   240 apply auto
```
```   241 done
```
```   242
```
```   243 lemma LIMSEQ_norm:
```
```   244   fixes a :: "'a::real_normed_vector"
```
```   245   shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
```
```   246 unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)
```
```   247
```
```   248 lemma LIMSEQ_ignore_initial_segment:
```
```   249   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   250 apply (rule metric_LIMSEQ_I)
```
```   251 apply (drule (1) metric_LIMSEQ_D)
```
```   252 apply (erule exE, rename_tac N)
```
```   253 apply (rule_tac x=N in exI)
```
```   254 apply simp
```
```   255 done
```
```   256
```
```   257 lemma LIMSEQ_offset:
```
```   258   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   259 apply (rule metric_LIMSEQ_I)
```
```   260 apply (drule (1) metric_LIMSEQ_D)
```
```   261 apply (erule exE, rename_tac N)
```
```   262 apply (rule_tac x="N + k" in exI)
```
```   263 apply clarify
```
```   264 apply (drule_tac x="n - k" in spec)
```
```   265 apply (simp add: le_diff_conv2)
```
```   266 done
```
```   267
```
```   268 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   269 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   270
```
```   271 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   272 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   273
```
```   274 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   275 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   276
```
```   277 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   278   unfolding LIMSEQ_def
```
```   279   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   280
```
```   281 lemma LIMSEQ_add:
```
```   282   fixes a b :: "'a::real_normed_vector"
```
```   283   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
```
```   284 unfolding LIMSEQ_conv_tendsto by (rule tendsto_add)
```
```   285
```
```   286 lemma LIMSEQ_minus:
```
```   287   fixes a :: "'a::real_normed_vector"
```
```   288   shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
```
```   289 unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)
```
```   290
```
```   291 lemma LIMSEQ_minus_cancel:
```
```   292   fixes a :: "'a::real_normed_vector"
```
```   293   shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
```
```   294 by (drule LIMSEQ_minus, simp)
```
```   295
```
```   296 lemma LIMSEQ_diff:
```
```   297   fixes a b :: "'a::real_normed_vector"
```
```   298   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
```
```   299 unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)
```
```   300
```
```   301 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   302 apply (rule ccontr)
```
```   303 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   304 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   305 apply (clarify, rename_tac M N)
```
```   306 apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
```
```   307 apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
```
```   308 apply (erule le_less_trans, rule add_strict_mono, simp, simp)
```
```   309 apply (subst dist_commute, rule dist_triangle)
```
```   310 done
```
```   311
```
```   312 lemma (in bounded_linear) LIMSEQ:
```
```   313   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```
```   314 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   315
```
```   316 lemma (in bounded_bilinear) LIMSEQ:
```
```   317   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
```
```   318 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   319
```
```   320 lemma LIMSEQ_mult:
```
```   321   fixes a b :: "'a::real_normed_algebra"
```
```   322   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
```
```   323 by (rule mult.LIMSEQ)
```
```   324
```
```   325 lemma increasing_LIMSEQ:
```
```   326   fixes f :: "nat \<Rightarrow> real"
```
```   327   assumes inc: "!!n. f n \<le> f (Suc n)"
```
```   328       and bdd: "!!n. f n \<le> l"
```
```   329       and en: "!!e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```   330   shows "f ----> l"
```
```   331 proof (auto simp add: LIMSEQ_def)
```
```   332   fix e :: real
```
```   333   assume e: "0 < e"
```
```   334   then obtain N where "l \<le> f N + e/2"
```
```   335     by (metis half_gt_zero e en that)
```
```   336   hence N: "l < f N + e" using e
```
```   337     by simp
```
```   338   { fix k
```
```   339     have [simp]: "!!n. \<bar>f n - l\<bar> = l - f n"
```
```   340       by (simp add: bdd)
```
```   341     have "\<bar>f (N+k) - l\<bar> < e"
```
```   342     proof (induct k)
```
```   343       case 0 show ?case using N
```
```   344         by simp
```
```   345     next
```
```   346       case (Suc k) thus ?case using N inc [of "N+k"]
```
```   347         by simp
```
```   348     qed
```
```   349   } note 1 = this
```
```   350   { fix n
```
```   351     have "N \<le> n \<Longrightarrow> \<bar>f n - l\<bar> < e" using 1 [of "n-N"]
```
```   352       by simp
```
```   353   } note [intro] = this
```
```   354   show " \<exists>no. \<forall>n\<ge>no. dist (f n) l < e"
```
```   355     by (auto simp add: dist_real_def)
```
```   356   qed
```
```   357
```
```   358 lemma Bseq_inverse_lemma:
```
```   359   fixes x :: "'a::real_normed_div_algebra"
```
```   360   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   361 apply (subst nonzero_norm_inverse, clarsimp)
```
```   362 apply (erule (1) le_imp_inverse_le)
```
```   363 done
```
```   364
```
```   365 lemma Bseq_inverse:
```
```   366   fixes a :: "'a::real_normed_div_algebra"
```
```   367   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```   368 unfolding LIMSEQ_conv_tendsto Bseq_conv_Bfun
```
```   369 by (rule Bfun_inverse)
```
```   370
```
```   371 lemma LIMSEQ_inverse:
```
```   372   fixes a :: "'a::real_normed_div_algebra"
```
```   373   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
```
```   374 unfolding LIMSEQ_conv_tendsto
```
```   375 by (rule tendsto_inverse)
```
```   376
```
```   377 lemma LIMSEQ_divide:
```
```   378   fixes a b :: "'a::real_normed_field"
```
```   379   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
```
```   380 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
```
```   381
```
```   382 lemma LIMSEQ_pow:
```
```   383   fixes a :: "'a::{power, real_normed_algebra}"
```
```   384   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
```
```   385 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
```
```   386
```
```   387 lemma LIMSEQ_setsum:
```
```   388   fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   389   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   390   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
```
```   391 using n unfolding LIMSEQ_conv_tendsto by (rule tendsto_setsum)
```
```   392
```
```   393 lemma LIMSEQ_setprod:
```
```   394   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
```
```   395   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   396   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
```
```   397 proof (cases "finite S")
```
```   398   case True
```
```   399   thus ?thesis using n
```
```   400   proof (induct)
```
```   401     case empty
```
```   402     show ?case
```
```   403       by (simp add: LIMSEQ_const)
```
```   404   next
```
```   405     case insert
```
```   406     thus ?case
```
```   407       by (simp add: LIMSEQ_mult)
```
```   408   qed
```
```   409 next
```
```   410   case False
```
```   411   thus ?thesis
```
```   412     by (simp add: setprod_def LIMSEQ_const)
```
```   413 qed
```
```   414
```
```   415 lemma LIMSEQ_add_const:
```
```   416   fixes a :: "'a::real_normed_vector"
```
```   417   shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
```
```   418 by (simp add: LIMSEQ_add LIMSEQ_const)
```
```   419
```
```   420 (* FIXME: delete *)
```
```   421 lemma LIMSEQ_add_minus:
```
```   422   fixes a b :: "'a::real_normed_vector"
```
```   423   shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
```
```   424 by (simp only: LIMSEQ_add LIMSEQ_minus)
```
```   425
```
```   426 lemma LIMSEQ_diff_const:
```
```   427   fixes a b :: "'a::real_normed_vector"
```
```   428   shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
```
```   429 by (simp add: LIMSEQ_diff LIMSEQ_const)
```
```   430
```
```   431 lemma LIMSEQ_diff_approach_zero:
```
```   432   fixes L :: "'a::real_normed_vector"
```
```   433   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```   434 by (drule (1) LIMSEQ_add, simp)
```
```   435
```
```   436 lemma LIMSEQ_diff_approach_zero2:
```
```   437   fixes L :: "'a::real_normed_vector"
```
```   438   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
```
```   439 by (drule (1) LIMSEQ_diff, simp)
```
```   440
```
```   441 text{*A sequence tends to zero iff its abs does*}
```
```   442 lemma LIMSEQ_norm_zero:
```
```   443   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   444   shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
```
```   445 by (simp add: LIMSEQ_iff)
```
```   446
```
```   447 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
```
```   448 by (simp add: LIMSEQ_iff)
```
```   449
```
```   450 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
```
```   451 by (drule LIMSEQ_norm, simp)
```
```   452
```
```   453 text{*An unbounded sequence's inverse tends to 0*}
```
```   454
```
```   455 lemma LIMSEQ_inverse_zero:
```
```   456   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   457 apply (rule LIMSEQ_I)
```
```   458 apply (drule_tac x="inverse r" in spec, safe)
```
```   459 apply (rule_tac x="N" in exI, safe)
```
```   460 apply (drule_tac x="n" in spec, safe)
```
```   461 apply (frule positive_imp_inverse_positive)
```
```   462 apply (frule (1) less_imp_inverse_less)
```
```   463 apply (subgoal_tac "0 < X n", simp)
```
```   464 apply (erule (1) order_less_trans)
```
```   465 done
```
```   466
```
```   467 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   468
```
```   469 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   470 apply (rule LIMSEQ_inverse_zero, safe)
```
```   471 apply (cut_tac x = r in reals_Archimedean2)
```
```   472 apply (safe, rule_tac x = n in exI)
```
```   473 apply (auto simp add: real_of_nat_Suc)
```
```   474 done
```
```   475
```
```   476 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   477 infinity is now easily proved*}
```
```   478
```
```   479 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   480      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   481 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   482
```
```   483 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   484      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   485 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   486
```
```   487 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   488      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   489 by (cut_tac b=1 in
```
```   490         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
```
```   491
```
```   492 lemma LIMSEQ_le_const:
```
```   493   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   494 apply (rule ccontr, simp only: linorder_not_le)
```
```   495 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
```
```   496 apply clarsimp
```
```   497 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
```
```   498 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
```
```   499 apply simp
```
```   500 done
```
```   501
```
```   502 lemma LIMSEQ_le_const2:
```
```   503   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   504 apply (subgoal_tac "- a \<le> - x", simp)
```
```   505 apply (rule LIMSEQ_le_const)
```
```   506 apply (erule LIMSEQ_minus)
```
```   507 apply simp
```
```   508 done
```
```   509
```
```   510 lemma LIMSEQ_le:
```
```   511   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
```
```   512 apply (subgoal_tac "0 \<le> y - x", simp)
```
```   513 apply (rule LIMSEQ_le_const)
```
```   514 apply (erule (1) LIMSEQ_diff)
```
```   515 apply (simp add: le_diff_eq)
```
```   516 done
```
```   517
```
```   518
```
```   519 subsection {* Convergence *}
```
```   520
```
```   521 lemma limI: "X ----> L ==> lim X = L"
```
```   522 apply (simp add: lim_def)
```
```   523 apply (blast intro: LIMSEQ_unique)
```
```   524 done
```
```   525
```
```   526 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   527 by (simp add: convergent_def)
```
```   528
```
```   529 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   530 by (auto simp add: convergent_def)
```
```   531
```
```   532 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   533 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   534
```
```   535 lemma convergent_const: "convergent (\<lambda>n. c)"
```
```   536 by (rule convergentI, rule LIMSEQ_const)
```
```   537
```
```   538 lemma convergent_add:
```
```   539   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   540   assumes "convergent (\<lambda>n. X n)"
```
```   541   assumes "convergent (\<lambda>n. Y n)"
```
```   542   shows "convergent (\<lambda>n. X n + Y n)"
```
```   543 using assms unfolding convergent_def by (fast intro: LIMSEQ_add)
```
```   544
```
```   545 lemma convergent_setsum:
```
```   546   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```   547   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```   548   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```   549 proof (cases "finite A")
```
```   550   case True from this and assms show ?thesis
```
```   551     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```   552 qed (simp add: convergent_const)
```
```   553
```
```   554 lemma (in bounded_linear) convergent:
```
```   555   assumes "convergent (\<lambda>n. X n)"
```
```   556   shows "convergent (\<lambda>n. f (X n))"
```
```   557 using assms unfolding convergent_def by (fast intro: LIMSEQ)
```
```   558
```
```   559 lemma (in bounded_bilinear) convergent:
```
```   560   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```   561   shows "convergent (\<lambda>n. X n ** Y n)"
```
```   562 using assms unfolding convergent_def by (fast intro: LIMSEQ)
```
```   563
```
```   564 lemma convergent_minus_iff:
```
```   565   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   566   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```   567 apply (simp add: convergent_def)
```
```   568 apply (auto dest: LIMSEQ_minus)
```
```   569 apply (drule LIMSEQ_minus, auto)
```
```   570 done
```
```   571
```
```   572 lemma lim_le:
```
```   573   fixes x :: real
```
```   574   assumes f: "convergent f" and fn_le: "!!n. f n \<le> x"
```
```   575   shows "lim f \<le> x"
```
```   576 proof (rule classical)
```
```   577   assume "\<not> lim f \<le> x"
```
```   578   hence 0: "0 < lim f - x" by arith
```
```   579   have 1: "f----> lim f"
```
```   580     by (metis convergent_LIMSEQ_iff f)
```
```   581   thus ?thesis
```
```   582     proof (simp add: LIMSEQ_iff)
```
```   583       assume "\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < r"
```
```   584       hence "\<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
```
```   585         by (metis 0)
```
```   586       from this obtain no where "\<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
```
```   587         by blast
```
```   588       thus "lim f \<le> x"
```
```   589         by (metis add_cancel_end add_minus_cancel diff_def linorder_linear
```
```   590                   linorder_not_le minus_diff_eq abs_diff_less_iff fn_le)
```
```   591     qed
```
```   592 qed
```
```   593
```
```   594 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   595
```
```   596 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   597   unfolding Ex1_def
```
```   598   apply (rule_tac x="nat_rec e f" in exI)
```
```   599   apply (rule conjI)+
```
```   600 apply (rule def_nat_rec_0, simp)
```
```   601 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   602 apply (rule allI, rule impI, rule ext)
```
```   603 apply (erule conjE)
```
```   604 apply (induct_tac x)
```
```   605 apply simp
```
```   606 apply (erule_tac x="n" in allE)
```
```   607 apply (simp)
```
```   608 done
```
```   609
```
```   610 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   611
```
```   612 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   613 apply (simp add: subseq_def)
```
```   614 apply (auto dest!: less_imp_Suc_add)
```
```   615 apply (induct_tac k)
```
```   616 apply (auto intro: less_trans)
```
```   617 done
```
```   618
```
```   619 lemma monoseq_Suc:
```
```   620    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
```
```   621                  | (\<forall>n. X (Suc n) \<le> X n))"
```
```   622 apply (simp add: monoseq_def)
```
```   623 apply (auto dest!: le_imp_less_or_eq)
```
```   624 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
```
```   625 apply (induct_tac "ka")
```
```   626 apply (auto intro: order_trans)
```
```   627 apply (erule contrapos_np)
```
```   628 apply (induct_tac "k")
```
```   629 apply (auto intro: order_trans)
```
```   630 done
```
```   631
```
```   632 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```   633 by (simp add: monoseq_def)
```
```   634
```
```   635 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```   636 by (simp add: monoseq_def)
```
```   637
```
```   638 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```   639 by (simp add: monoseq_Suc)
```
```   640
```
```   641 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```   642 by (simp add: monoseq_Suc)
```
```   643
```
```   644 lemma monoseq_minus: assumes "monoseq a"
```
```   645   shows "monoseq (\<lambda> n. - a n)"
```
```   646 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   647   case True
```
```   648   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   649   thus ?thesis by (rule monoI2)
```
```   650 next
```
```   651   case False
```
```   652   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   653   thus ?thesis by (rule monoI1)
```
```   654 qed
```
```   655
```
```   656 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
```
```   657   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
```
```   658          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   659 proof -
```
```   660   { fix x n fix a :: "nat \<Rightarrow> real"
```
```   661     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
```
```   662     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
```
```   663     have "a n \<le> x"
```
```   664     proof (rule ccontr)
```
```   665       assume "\<not> a n \<le> x" hence "x < a n" by auto
```
```   666       hence "0 < a n - x" by auto
```
```   667       from `a ----> x`[THEN LIMSEQ_D, OF this]
```
```   668       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
```
```   669       hence "norm (a (max no n) - x) < a n - x" by auto
```
```   670       moreover
```
```   671       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
```
```   672       hence "x < a (max no n)" by auto
```
```   673       ultimately
```
```   674       have "a (max no n) < a n" by auto
```
```   675       with monotone[where m=n and n="max no n"]
```
```   676       show False by (auto simp:max_def split:split_if_asm)
```
```   677     qed
```
```   678   } note top_down = this
```
```   679   { fix x n m fix a :: "nat \<Rightarrow> real"
```
```   680     assume "a ----> x" and "monoseq a" and "a m < x"
```
```   681     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
```
```   682     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   683       case True with top_down and `a ----> x` show ?thesis by auto
```
```   684     next
```
```   685       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
```
```   686       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
```
```   687       hence False using `a m < x` by auto
```
```   688       thus ?thesis ..
```
```   689     qed
```
```   690   } note when_decided = this
```
```   691
```
```   692   show ?thesis
```
```   693   proof (cases "\<exists> m. a m \<noteq> x")
```
```   694     case True then obtain m where "a m \<noteq> x" by auto
```
```   695     show ?thesis
```
```   696     proof (cases "a m < x")
```
```   697       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
```
```   698       show ?thesis by blast
```
```   699     next
```
```   700       case False hence "- a m < - x" using `a m \<noteq> x` by auto
```
```   701       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
```
```   702       show ?thesis by auto
```
```   703     qed
```
```   704   qed auto
```
```   705 qed
```
```   706
```
```   707 text{* for any sequence, there is a mootonic subsequence *}
```
```   708 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   709 proof-
```
```   710   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   711     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   712     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   713     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   714     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   715       using H apply -
```
```   716       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   717       unfolding order_le_less by blast
```
```   718     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   719     {fix n
```
```   720       have "?P (f (Suc n)) (f n)"
```
```   721         unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   722         using H apply -
```
```   723       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   724       unfolding order_le_less by blast
```
```   725     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   726   note fSuc = this
```
```   727     {fix p q assume pq: "p \<ge> f q"
```
```   728       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   729         by (cases q, simp_all) }
```
```   730     note pqth = this
```
```   731     {fix q
```
```   732       have "f (Suc q) > f q" apply (induct q)
```
```   733         using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   734     note fss = this
```
```   735     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   736     {fix a b
```
```   737       have "f a \<le> f (a + b)"
```
```   738       proof(induct b)
```
```   739         case 0 thus ?case by simp
```
```   740       next
```
```   741         case (Suc b)
```
```   742         from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   743       qed}
```
```   744     note fmon0 = this
```
```   745     have "monoseq (\<lambda>n. s (f n))"
```
```   746     proof-
```
```   747       {fix n
```
```   748         have "s (f n) \<ge> s (f (Suc n))"
```
```   749         proof(cases n)
```
```   750           case 0
```
```   751           assume n0: "n = 0"
```
```   752           from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   753           from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   754         next
```
```   755           case (Suc m)
```
```   756           assume m: "n = Suc m"
```
```   757           from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   758           from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   759         qed}
```
```   760       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   761     qed
```
```   762     with th1 have ?thesis by blast}
```
```   763   moreover
```
```   764   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   765     {fix p assume p: "p \<ge> Suc N"
```
```   766       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   767       have "m \<noteq> p" using m(2) by auto
```
```   768       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   769     note th0 = this
```
```   770     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   771     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   772     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   773       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   774     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   775       using N apply -
```
```   776       apply (erule allE[where x="Suc N"], clarsimp)
```
```   777       apply (rule_tac x="m" in exI)
```
```   778       apply auto
```
```   779       apply (subgoal_tac "Suc N \<noteq> m")
```
```   780       apply simp
```
```   781       apply (rule ccontr, simp)
```
```   782       done
```
```   783     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   784     {fix n
```
```   785       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   786         unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   787       proof (induct n)
```
```   788         case 0 thus ?case
```
```   789           using f0 N apply auto
```
```   790           apply (erule allE[where x="f 0"], clarsimp)
```
```   791           apply (rule_tac x="m" in exI, simp)
```
```   792           by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   793       next
```
```   794         case (Suc n)
```
```   795         from Suc.hyps have Nfn: "N < f n" by blast
```
```   796         from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   797         with Nfn have mN: "m > N" by arith
```
```   798         note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   799
```
```   800         from key have th0: "f (Suc n) > N" by simp
```
```   801         from N[rule_format, OF th0]
```
```   802         obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   803         have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   804         hence "m' > f (Suc n)" using m'(1) by simp
```
```   805         with key m'(2) show ?case by auto
```
```   806       qed}
```
```   807     note fSuc = this
```
```   808     {fix n
```
```   809       have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto
```
```   810       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   811     note thf = this
```
```   812     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   813     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   814       apply -
```
```   815       apply (rule disjI1)
```
```   816       apply auto
```
```   817       apply (rule order_less_imp_le)
```
```   818       apply blast
```
```   819       done
```
```   820     then have ?thesis  using sqf by blast}
```
```   821   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   822 qed
```
```   823
```
```   824 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   825 proof(induct n)
```
```   826   case 0 thus ?case by simp
```
```   827 next
```
```   828   case (Suc n)
```
```   829   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   830   have "n < f (Suc n)" by arith
```
```   831   thus ?case by arith
```
```   832 qed
```
```   833
```
```   834 lemma LIMSEQ_subseq_LIMSEQ:
```
```   835   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```   836 apply (auto simp add: LIMSEQ_def)
```
```   837 apply (drule_tac x=r in spec, clarify)
```
```   838 apply (rule_tac x=no in exI, clarify)
```
```   839 apply (blast intro: seq_suble le_trans dest!: spec)
```
```   840 done
```
```   841
```
```   842 subsection {* Bounded Monotonic Sequences *}
```
```   843
```
```   844
```
```   845 text{*Bounded Sequence*}
```
```   846
```
```   847 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   848 by (simp add: Bseq_def)
```
```   849
```
```   850 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   851 by (auto simp add: Bseq_def)
```
```   852
```
```   853 lemma lemma_NBseq_def:
```
```   854      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
```
```   855       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   856 proof auto
```
```   857   fix K :: real
```
```   858   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```   859   then have "K \<le> real (Suc n)" by auto
```
```   860   assume "\<forall>m. norm (X m) \<le> K"
```
```   861   have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```   862   proof
```
```   863     fix m :: 'a
```
```   864     from `\<forall>m. norm (X m) \<le> K` have "norm (X m) \<le> K" ..
```
```   865     with `K \<le> real (Suc n)` show "norm (X m) \<le> real (Suc n)" by auto
```
```   866   qed
```
```   867   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```   868 next
```
```   869   fix N :: nat
```
```   870   have "real (Suc N) > 0" by (simp add: real_of_nat_Suc)
```
```   871   moreover assume "\<forall>n. norm (X n) \<le> real (Suc N)"
```
```   872   ultimately show "\<exists>K>0. \<forall>n. norm (X n) \<le> K" by blast
```
```   873 qed
```
```   874
```
```   875
```
```   876 text{* alternative definition for Bseq *}
```
```   877 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   878 apply (simp add: Bseq_def)
```
```   879 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   880 done
```
```   881
```
```   882 lemma lemma_NBseq_def2:
```
```   883      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   884 apply (subst lemma_NBseq_def, auto)
```
```   885 apply (rule_tac x = "Suc N" in exI)
```
```   886 apply (rule_tac [2] x = N in exI)
```
```   887 apply (auto simp add: real_of_nat_Suc)
```
```   888  prefer 2 apply (blast intro: order_less_imp_le)
```
```   889 apply (drule_tac x = n in spec, simp)
```
```   890 done
```
```   891
```
```   892 (* yet another definition for Bseq *)
```
```   893 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   894 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   895
```
```   896 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   897
```
```   898 lemma Bseq_isUb:
```
```   899   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   900 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   901
```
```   902
```
```   903 text{* Use completeness of reals (supremum property)
```
```   904    to show that any bounded sequence has a least upper bound*}
```
```   905
```
```   906 lemma Bseq_isLub:
```
```   907   "!!(X::nat=>real). Bseq X ==>
```
```   908    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   909 by (blast intro: reals_complete Bseq_isUb)
```
```   910
```
```   911 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   912
```
```   913 lemma lemma_converg1:
```
```   914      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
```
```   915                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
```
```   916                |] ==> \<forall>n \<ge> ma. X n = X ma"
```
```   917 apply safe
```
```   918 apply (drule_tac y = "X n" in isLubD2)
```
```   919 apply (blast dest: order_antisym)+
```
```   920 done
```
```   921
```
```   922 text{* The best of both worlds: Easier to prove this result as a standard
```
```   923    theorem and then use equivalence to "transfer" it into the
```
```   924    equivalent nonstandard form if needed!*}
```
```   925
```
```   926 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   927 apply (simp add: LIMSEQ_def)
```
```   928 apply (rule_tac x = "X m" in exI, safe)
```
```   929 apply (rule_tac x = m in exI, safe)
```
```   930 apply (drule spec, erule impE, auto)
```
```   931 done
```
```   932
```
```   933 lemma lemma_converg2:
```
```   934    "!!(X::nat=>real).
```
```   935     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
```
```   936 apply safe
```
```   937 apply (drule_tac y = "X m" in isLubD2)
```
```   938 apply (auto dest!: order_le_imp_less_or_eq)
```
```   939 done
```
```   940
```
```   941 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
```
```   942 by (rule setleI [THEN isUbI], auto)
```
```   943
```
```   944 text{* FIXME: @{term "U - T < U"} is redundant *}
```
```   945 lemma lemma_converg4: "!!(X::nat=> real).
```
```   946                [| \<forall>m. X m ~= U;
```
```   947                   isLub UNIV {x. \<exists>n. X n = x} U;
```
```   948                   0 < T;
```
```   949                   U + - T < U
```
```   950                |] ==> \<exists>m. U + -T < X m & X m < U"
```
```   951 apply (drule lemma_converg2, assumption)
```
```   952 apply (rule ccontr, simp)
```
```   953 apply (simp add: linorder_not_less)
```
```   954 apply (drule lemma_converg3)
```
```   955 apply (drule isLub_le_isUb, assumption)
```
```   956 apply (auto dest: order_less_le_trans)
```
```   957 done
```
```   958
```
```   959 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```   960
```
```   961 lemma Bseq_mono_convergent:
```
```   962      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
```
```   963 apply (simp add: convergent_def)
```
```   964 apply (frule Bseq_isLub, safe)
```
```   965 apply (case_tac "\<exists>m. X m = U", auto)
```
```   966 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
```
```   967 (* second case *)
```
```   968 apply (rule_tac x = U in exI)
```
```   969 apply (subst LIMSEQ_iff, safe)
```
```   970 apply (frule lemma_converg2, assumption)
```
```   971 apply (drule lemma_converg4, auto)
```
```   972 apply (rule_tac x = m in exI, safe)
```
```   973 apply (subgoal_tac "X m \<le> X n")
```
```   974  prefer 2 apply blast
```
```   975 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
```
```   976 done
```
```   977
```
```   978 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```   979 by (simp add: Bseq_def)
```
```   980
```
```   981 text{*Main monotonicity theorem*}
```
```   982 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
```
```   983 apply (simp add: monoseq_def, safe)
```
```   984 apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
```
```   985 apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
```
```   986 apply (auto intro!: Bseq_mono_convergent)
```
```   987 done
```
```   988
```
```   989 subsubsection{*Increasing and Decreasing Series*}
```
```   990
```
```   991 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```   992   by (simp add: incseq_def monoseq_def)
```
```   993
```
```   994 lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
```
```   995   using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
```
```   996 proof
```
```   997   assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
```
```   998   thus ?thesis by simp
```
```   999 next
```
```  1000   assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
```
```  1001   hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
```
```  1002     by (auto simp add: incseq_def intro: order_antisym)
```
```  1003   have X: "!!n. X n = X 0"
```
```  1004     by (blast intro: const [of 0])
```
```  1005   have "X = (\<lambda>n. X 0)"
```
```  1006     by (blast intro: ext X)
```
```  1007   hence "L = X 0" using LIMSEQ_const [of "X 0"]
```
```  1008     by (auto intro: LIMSEQ_unique lim)
```
```  1009   thus ?thesis
```
```  1010     by (blast intro: eq_refl X)
```
```  1011 qed
```
```  1012
```
```  1013 lemma incseq_SucI:
```
```  1014   assumes "\<And>n. X n \<le> X (Suc n)"
```
```  1015   shows "incseq X" unfolding incseq_def
```
```  1016 proof safe
```
```  1017   fix m n :: nat
```
```  1018   { fix d m :: nat
```
```  1019     have "X m \<le> X (m + d)"
```
```  1020     proof (induct d)
```
```  1021       case (Suc d)
```
```  1022       also have "X (m + d) \<le> X (m + Suc d)"
```
```  1023         using assms by simp
```
```  1024       finally show ?case .
```
```  1025     qed simp }
```
```  1026   note this[of m "n - m"]
```
```  1027   moreover assume "m \<le> n"
```
```  1028   ultimately show "X m \<le> X n" by simp
```
```  1029 qed
```
```  1030
```
```  1031 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```  1032   by (simp add: decseq_def monoseq_def)
```
```  1033
```
```  1034 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
```
```  1035   by (simp add: decseq_def incseq_def)
```
```  1036
```
```  1037
```
```  1038 lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
```
```  1039 proof -
```
```  1040   have inc: "incseq (\<lambda>n. - X n)" using dec
```
```  1041     by (simp add: decseq_eq_incseq)
```
```  1042   have "- X n \<le> - L"
```
```  1043     by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim)
```
```  1044   thus ?thesis
```
```  1045     by simp
```
```  1046 qed
```
```  1047
```
```  1048 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```  1049
```
```  1050 text{*alternative formulation for boundedness*}
```
```  1051 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```  1052 apply (unfold Bseq_def, safe)
```
```  1053 apply (rule_tac [2] x = "k + norm x" in exI)
```
```  1054 apply (rule_tac x = K in exI, simp)
```
```  1055 apply (rule exI [where x = 0], auto)
```
```  1056 apply (erule order_less_le_trans, simp)
```
```  1057 apply (drule_tac x=n in spec, fold diff_def)
```
```  1058 apply (drule order_trans [OF norm_triangle_ineq2])
```
```  1059 apply simp
```
```  1060 done
```
```  1061
```
```  1062 text{*alternative formulation for boundedness*}
```
```  1063 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```  1064 apply safe
```
```  1065 apply (simp add: Bseq_def, safe)
```
```  1066 apply (rule_tac x = "K + norm (X N)" in exI)
```
```  1067 apply auto
```
```  1068 apply (erule order_less_le_trans, simp)
```
```  1069 apply (rule_tac x = N in exI, safe)
```
```  1070 apply (drule_tac x = n in spec)
```
```  1071 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```  1072 apply (auto simp add: Bseq_iff2)
```
```  1073 done
```
```  1074
```
```  1075 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```  1076 apply (simp add: Bseq_def)
```
```  1077 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```  1078 apply (drule_tac x = n in spec, arith)
```
```  1079 done
```
```  1080
```
```  1081
```
```  1082 subsection {* Cauchy Sequences *}
```
```  1083
```
```  1084 lemma metric_CauchyI:
```
```  1085   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1086 by (simp add: Cauchy_def)
```
```  1087
```
```  1088 lemma metric_CauchyD:
```
```  1089   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```  1090 by (simp add: Cauchy_def)
```
```  1091
```
```  1092 lemma Cauchy_iff:
```
```  1093   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1094   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1095 unfolding Cauchy_def dist_norm ..
```
```  1096
```
```  1097 lemma Cauchy_iff2:
```
```  1098      "Cauchy X =
```
```  1099       (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```  1100 apply (simp add: Cauchy_iff, auto)
```
```  1101 apply (drule reals_Archimedean, safe)
```
```  1102 apply (drule_tac x = n in spec, auto)
```
```  1103 apply (rule_tac x = M in exI, auto)
```
```  1104 apply (drule_tac x = m in spec, simp)
```
```  1105 apply (drule_tac x = na in spec, auto)
```
```  1106 done
```
```  1107
```
```  1108 lemma CauchyI:
```
```  1109   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1110   shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
```
```  1111 by (simp add: Cauchy_iff)
```
```  1112
```
```  1113 lemma CauchyD:
```
```  1114   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1115   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1116 by (simp add: Cauchy_iff)
```
```  1117
```
```  1118 lemma Cauchy_subseq_Cauchy:
```
```  1119   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```  1120 apply (auto simp add: Cauchy_def)
```
```  1121 apply (drule_tac x=e in spec, clarify)
```
```  1122 apply (rule_tac x=M in exI, clarify)
```
```  1123 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```  1124 done
```
```  1125
```
```  1126 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1127
```
```  1128 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1129   proof mechanization rather than the nonstandard proof*}
```
```  1130
```
```  1131 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1132           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1133 apply (clarify, drule spec, drule (1) mp)
```
```  1134 apply (simp only: norm_minus_commute)
```
```  1135 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1136 apply simp
```
```  1137 done
```
```  1138
```
```  1139 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```  1140 apply (simp add: Cauchy_iff)
```
```  1141 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```  1142 apply (drule_tac x="M" in spec, simp)
```
```  1143 apply (drule lemmaCauchy)
```
```  1144 apply (rule_tac k="M" in Bseq_offset)
```
```  1145 apply (simp add: Bseq_def)
```
```  1146 apply (rule_tac x="1 + norm (X M)" in exI)
```
```  1147 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```  1148 apply (simp add: order_less_imp_le)
```
```  1149 done
```
```  1150
```
```  1151 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1152
```
```  1153 class complete_space =
```
```  1154   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1155
```
```  1156 class banach = real_normed_vector + complete_space
```
```  1157
```
```  1158 theorem LIMSEQ_imp_Cauchy:
```
```  1159   assumes X: "X ----> a" shows "Cauchy X"
```
```  1160 proof (rule metric_CauchyI)
```
```  1161   fix e::real assume "0 < e"
```
```  1162   hence "0 < e/2" by simp
```
```  1163   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```  1164   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```  1165   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```  1166   proof (intro exI allI impI)
```
```  1167     fix m assume "N \<le> m"
```
```  1168     hence m: "dist (X m) a < e/2" using N by fast
```
```  1169     fix n assume "N \<le> n"
```
```  1170     hence n: "dist (X n) a < e/2" using N by fast
```
```  1171     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```  1172       by (rule dist_triangle2)
```
```  1173     also from m n have "\<dots> < e" by simp
```
```  1174     finally show "dist (X m) (X n) < e" .
```
```  1175   qed
```
```  1176 qed
```
```  1177
```
```  1178 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1179 unfolding convergent_def
```
```  1180 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1181
```
```  1182 lemma Cauchy_convergent_iff:
```
```  1183   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1184   shows "Cauchy X = convergent X"
```
```  1185 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1186
```
```  1187 lemma convergent_subseq_convergent:
```
```  1188   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1189   shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```  1190   by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
```
```  1191
```
```  1192 text {*
```
```  1193 Proof that Cauchy sequences converge based on the one from
```
```  1194 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1195 *}
```
```  1196
```
```  1197 text {*
```
```  1198   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1199   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1200 *}
```
```  1201
```
```  1202 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1203 by (simp add: isUbI setleI)
```
```  1204
```
```  1205 locale real_Cauchy =
```
```  1206   fixes X :: "nat \<Rightarrow> real"
```
```  1207   assumes X: "Cauchy X"
```
```  1208   fixes S :: "real set"
```
```  1209   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1210
```
```  1211 lemma real_CauchyI:
```
```  1212   assumes "Cauchy X"
```
```  1213   shows "real_Cauchy X"
```
```  1214   proof qed (fact assms)
```
```  1215
```
```  1216 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
```
```  1217 by (unfold S_def, auto)
```
```  1218
```
```  1219 lemma (in real_Cauchy) bound_isUb:
```
```  1220   assumes N: "\<forall>n\<ge>N. X n < x"
```
```  1221   shows "isUb UNIV S x"
```
```  1222 proof (rule isUb_UNIV_I)
```
```  1223   fix y::real assume "y \<in> S"
```
```  1224   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1225     by (simp add: S_def)
```
```  1226   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1227   hence "y < X (max M N)" by simp
```
```  1228   also have "\<dots> < x" using N by simp
```
```  1229   finally show "y \<le> x"
```
```  1230     by (rule order_less_imp_le)
```
```  1231 qed
```
```  1232
```
```  1233 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
```
```  1234 proof (rule reals_complete)
```
```  1235   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```  1236     using CauchyD [OF X zero_less_one] by auto
```
```  1237   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```  1238   show "\<exists>x. x \<in> S"
```
```  1239   proof
```
```  1240     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1241       by (simp add: abs_diff_less_iff)
```
```  1242     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1243   qed
```
```  1244   show "\<exists>u. isUb UNIV S u"
```
```  1245   proof
```
```  1246     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1247       by (simp add: abs_diff_less_iff)
```
```  1248     thus "isUb UNIV S (X N + 1)"
```
```  1249       by (rule bound_isUb)
```
```  1250   qed
```
```  1251 qed
```
```  1252
```
```  1253 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
```
```  1254   assumes x: "isLub UNIV S x"
```
```  1255   shows "X ----> x"
```
```  1256 proof (rule LIMSEQ_I)
```
```  1257   fix r::real assume "0 < r"
```
```  1258   hence r: "0 < r/2" by simp
```
```  1259   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```  1260     using CauchyD [OF X r] by auto
```
```  1261   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```  1262   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1263     by (simp only: real_norm_def abs_diff_less_iff)
```
```  1264
```
```  1265   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1266   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1267   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1268
```
```  1269   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1270   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1271   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1272
```
```  1273   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```  1274   proof (intro exI allI impI)
```
```  1275     fix n assume n: "N \<le> n"
```
```  1276     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1277     thus "norm (X n - x) < r" using 1 2
```
```  1278       by (simp add: abs_diff_less_iff)
```
```  1279   qed
```
```  1280 qed
```
```  1281
```
```  1282 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
```
```  1283 proof -
```
```  1284   obtain x where "isLub UNIV S x"
```
```  1285     using isLub_ex by fast
```
```  1286   hence "X ----> x"
```
```  1287     by (rule isLub_imp_LIMSEQ)
```
```  1288   thus ?thesis ..
```
```  1289 qed
```
```  1290
```
```  1291 lemma real_Cauchy_convergent:
```
```  1292   fixes X :: "nat \<Rightarrow> real"
```
```  1293   shows "Cauchy X \<Longrightarrow> convergent X"
```
```  1294 unfolding convergent_def
```
```  1295 by (rule real_Cauchy.LIMSEQ_ex)
```
```  1296  (rule real_CauchyI)
```
```  1297
```
```  1298 instance real :: banach
```
```  1299 by intro_classes (rule real_Cauchy_convergent)
```
```  1300
```
```  1301
```
```  1302 subsection {* Power Sequences *}
```
```  1303
```
```  1304 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1305 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1306   also fact that bounded and monotonic sequence converges.*}
```
```  1307
```
```  1308 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1309 apply (simp add: Bseq_def)
```
```  1310 apply (rule_tac x = 1 in exI)
```
```  1311 apply (simp add: power_abs)
```
```  1312 apply (auto dest: power_mono)
```
```  1313 done
```
```  1314
```
```  1315 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1316 apply (clarify intro!: mono_SucI2)
```
```  1317 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1318 done
```
```  1319
```
```  1320 lemma convergent_realpow:
```
```  1321   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1322 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1323
```
```  1324 lemma LIMSEQ_inverse_realpow_zero_lemma:
```
```  1325   fixes x :: real
```
```  1326   assumes x: "0 \<le> x"
```
```  1327   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1328 apply (induct n)
```
```  1329 apply simp
```
```  1330 apply simp
```
```  1331 apply (rule order_trans)
```
```  1332 prefer 2
```
```  1333 apply (erule mult_left_mono)
```
```  1334 apply (rule add_increasing [OF x], simp)
```
```  1335 apply (simp add: real_of_nat_Suc)
```
```  1336 apply (simp add: ring_distribs)
```
```  1337 apply (simp add: mult_nonneg_nonneg x)
```
```  1338 done
```
```  1339
```
```  1340 lemma LIMSEQ_inverse_realpow_zero:
```
```  1341   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1342 proof (rule LIMSEQ_inverse_zero [rule_format])
```
```  1343   fix y :: real
```
```  1344   assume x: "1 < x"
```
```  1345   hence "0 < x - 1" by simp
```
```  1346   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
```
```  1347     by (rule reals_Archimedean3)
```
```  1348   hence "\<exists>N::nat. y < real N * (x - 1)" ..
```
```  1349   then obtain N::nat where "y < real N * (x - 1)" ..
```
```  1350   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
```
```  1351   also have "\<dots> \<le> (x - 1 + 1) ^ N"
```
```  1352     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
```
```  1353   also have "\<dots> = x ^ N" by simp
```
```  1354   finally have "y < x ^ N" .
```
```  1355   hence "\<forall>n\<ge>N. y < x ^ n"
```
```  1356     apply clarify
```
```  1357     apply (erule order_less_le_trans)
```
```  1358     apply (erule power_increasing)
```
```  1359     apply (rule order_less_imp_le [OF x])
```
```  1360     done
```
```  1361   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
```
```  1362 qed
```
```  1363
```
```  1364 lemma LIMSEQ_realpow_zero:
```
```  1365   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1366 proof (cases)
```
```  1367   assume "x = 0"
```
```  1368   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
```
```  1369   thus ?thesis by (rule LIMSEQ_imp_Suc)
```
```  1370 next
```
```  1371   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1372   hence x0: "0 < x" by simp
```
```  1373   assume x1: "x < 1"
```
```  1374   from x0 x1 have "1 < inverse x"
```
```  1375     by (rule real_inverse_gt_one)
```
```  1376   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1377     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1378   thus ?thesis by (simp add: power_inverse)
```
```  1379 qed
```
```  1380
```
```  1381 lemma LIMSEQ_power_zero:
```
```  1382   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1383   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1384 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1385 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
```
```  1386 apply (simp add: power_abs norm_power_ineq)
```
```  1387 done
```
```  1388
```
```  1389 lemma LIMSEQ_divide_realpow_zero:
```
```  1390   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
```
```  1391 apply (cut_tac a = a and x1 = "inverse x" in
```
```  1392         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
```
```  1393 apply (auto simp add: divide_inverse power_inverse)
```
```  1394 apply (simp add: inverse_eq_divide pos_divide_less_eq)
```
```  1395 done
```
```  1396
```
```  1397 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1398
```
```  1399 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
```
```  1400 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1401
```
```  1402 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
```
```  1403 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
```
```  1404 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
```
```  1405 done
```
```  1406
```
```  1407 end
```