src/HOL/SEQ.thy
 author huffman Sun May 31 21:59:33 2009 -0700 (2009-05-31) changeset 31349 2261c8781f73 parent 31336 e17f13cd1280 child 31353 14a58e2ca374 permissions -rw-r--r--
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
```     1 (*  Title       : SEQ.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Description : Convergence of sequences and series
```
```     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     6     Additional contributions by Jeremy Avigad and Brian Huffman
```
```     7 *)
```
```     8
```
```     9 header {* Sequences and Convergence *}
```
```    10
```
```    11 theory SEQ
```
```    12 imports Limits
```
```    13 begin
```
```    14
```
```    15 definition
```
```    16   sequentially :: "nat filter" where
```
```    17   "sequentially = Abs_filter (\<lambda>P. \<exists>N. \<forall>n\<ge>N. P n)"
```
```    18
```
```    19 definition
```
```    20   Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
```
```    21     --{*Standard definition of sequence converging to zero*}
```
```    22   [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
```
```    23
```
```    24 definition
```
```    25   LIMSEQ :: "[nat \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
```
```    26     ("((_)/ ----> (_))" [60, 60] 60) where
```
```    27     --{*Standard definition of convergence of sequence*}
```
```    28   [code del]: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```    29
```
```    30 definition
```
```    31   lim :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where
```
```    32     --{*Standard definition of limit using choice operator*}
```
```    33   "lim X = (THE L. X ----> L)"
```
```    34
```
```    35 definition
```
```    36   convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    37     --{*Standard definition of convergence*}
```
```    38   "convergent X = (\<exists>L. X ----> L)"
```
```    39
```
```    40 definition
```
```    41   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
```
```    42     --{*Standard definition for bounded sequence*}
```
```    43   [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
```
```    44
```
```    45 definition
```
```    46   monoseq :: "(nat=>real)=>bool" where
```
```    47     --{*Definition of monotonicity.
```
```    48         The use of disjunction here complicates proofs considerably.
```
```    49         One alternative is to add a Boolean argument to indicate the direction.
```
```    50         Another is to develop the notions of increasing and decreasing first.*}
```
```    51   [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))"
```
```    52
```
```    53 definition
```
```    54   incseq :: "(nat=>real)=>bool" where
```
```    55     --{*Increasing sequence*}
```
```    56   [code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
```
```    57
```
```    58 definition
```
```    59   decseq :: "(nat=>real)=>bool" where
```
```    60     --{*Increasing sequence*}
```
```    61   [code del]: "decseq X = (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```    62
```
```    63 definition
```
```    64   subseq :: "(nat => nat) => bool" where
```
```    65     --{*Definition of subsequence*}
```
```    66   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
```
```    67
```
```    68 definition
```
```    69   Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    70     --{*Standard definition of the Cauchy condition*}
```
```    71   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```    72
```
```    73
```
```    74 subsection {* Sequentially *}
```
```    75
```
```    76 lemma eventually_sequentially:
```
```    77   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```    78 unfolding sequentially_def
```
```    79 apply (rule eventually_Abs_filter)
```
```    80 apply simp
```
```    81 apply (clarify, rule_tac x=N in exI, simp)
```
```    82 apply (clarify, rename_tac M N)
```
```    83 apply (rule_tac x="max M N" in exI, simp)
```
```    84 done
```
```    85
```
```    86 lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
```
```    87 unfolding Zseq_def Zfun_def eventually_sequentially ..
```
```    88
```
```    89 lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
```
```    90 unfolding LIMSEQ_def tendsto_def eventually_sequentially ..
```
```    91
```
```    92 subsection {* Bounded Sequences *}
```
```    93
```
```    94 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
```
```    95 unfolding Bseq_def
```
```    96 proof (intro exI conjI allI)
```
```    97   show "0 < max K 1" by simp
```
```    98 next
```
```    99   fix n::nat
```
```   100   have "norm (X n) \<le> K" by (rule K)
```
```   101   thus "norm (X n) \<le> max K 1" by simp
```
```   102 qed
```
```   103
```
```   104 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   105 unfolding Bseq_def by auto
```
```   106
```
```   107 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
```
```   108 proof (rule BseqI')
```
```   109   let ?A = "norm ` X ` {..N}"
```
```   110   have 1: "finite ?A" by simp
```
```   111   fix n::nat
```
```   112   show "norm (X n) \<le> max K (Max ?A)"
```
```   113   proof (cases rule: linorder_le_cases)
```
```   114     assume "n \<ge> N"
```
```   115     hence "norm (X n) \<le> K" using K by simp
```
```   116     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   117   next
```
```   118     assume "n \<le> N"
```
```   119     hence "norm (X n) \<in> ?A" by simp
```
```   120     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
```
```   121     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   122   qed
```
```   123 qed
```
```   124
```
```   125 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```   126 unfolding Bseq_def by auto
```
```   127
```
```   128 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```   129 apply (erule BseqE)
```
```   130 apply (rule_tac N="k" and K="K" in BseqI2')
```
```   131 apply clarify
```
```   132 apply (drule_tac x="n - k" in spec, simp)
```
```   133 done
```
```   134
```
```   135
```
```   136 subsection {* Sequences That Converge to Zero *}
```
```   137
```
```   138 lemma ZseqI:
```
```   139   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
```
```   140 unfolding Zseq_def by simp
```
```   141
```
```   142 lemma ZseqD:
```
```   143   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
```
```   144 unfolding Zseq_def by simp
```
```   145
```
```   146 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
```
```   147 unfolding Zseq_def by simp
```
```   148
```
```   149 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
```
```   150 unfolding Zseq_def by force
```
```   151
```
```   152 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
```
```   153 unfolding Zseq_def by simp
```
```   154
```
```   155 lemma Zseq_imp_Zseq:
```
```   156   assumes X: "Zseq X"
```
```   157   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
```
```   158   shows "Zseq (\<lambda>n. Y n)"
```
```   159 using assms unfolding Zseq_conv_Zfun by (rule Zfun_imp_Zfun)
```
```   160
```
```   161 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
```
```   162 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
```
```   163
```
```   164 lemma Zseq_add:
```
```   165   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"
```
```   166 unfolding Zseq_conv_Zfun by (rule Zfun_add)
```
```   167
```
```   168 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
```
```   169 unfolding Zseq_def by simp
```
```   170
```
```   171 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
```
```   172 by (simp only: diff_minus Zseq_add Zseq_minus)
```
```   173
```
```   174 lemma (in bounded_linear) Zseq:
```
```   175   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
```
```   176 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   177
```
```   178 lemma (in bounded_bilinear) Zseq:
```
```   179   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   180 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   181
```
```   182 lemma (in bounded_bilinear) Zseq_prod_Bseq:
```
```   183   assumes X: "Zseq X"
```
```   184   assumes Y: "Bseq Y"
```
```   185   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   186 proof -
```
```   187   obtain K where K: "0 \<le> K"
```
```   188     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   189     using nonneg_bounded by fast
```
```   190   obtain B where B: "0 < B"
```
```   191     and norm_Y: "\<And>n. norm (Y n) \<le> B"
```
```   192     using Y [unfolded Bseq_def] by fast
```
```   193   from X show ?thesis
```
```   194   proof (rule Zseq_imp_Zseq)
```
```   195     fix n::nat
```
```   196     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   197       by (rule norm_le)
```
```   198     also have "\<dots> \<le> norm (X n) * B * K"
```
```   199       by (intro mult_mono' order_refl norm_Y norm_ge_zero
```
```   200                 mult_nonneg_nonneg K)
```
```   201     also have "\<dots> = norm (X n) * (B * K)"
```
```   202       by (rule mult_assoc)
```
```   203     finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
```
```   204   qed
```
```   205 qed
```
```   206
```
```   207 lemma (in bounded_bilinear) Bseq_prod_Zseq:
```
```   208   assumes X: "Bseq X"
```
```   209   assumes Y: "Zseq Y"
```
```   210   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   211 proof -
```
```   212   obtain K where K: "0 \<le> K"
```
```   213     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   214     using nonneg_bounded by fast
```
```   215   obtain B where B: "0 < B"
```
```   216     and norm_X: "\<And>n. norm (X n) \<le> B"
```
```   217     using X [unfolded Bseq_def] by fast
```
```   218   from Y show ?thesis
```
```   219   proof (rule Zseq_imp_Zseq)
```
```   220     fix n::nat
```
```   221     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   222       by (rule norm_le)
```
```   223     also have "\<dots> \<le> B * norm (Y n) * K"
```
```   224       by (intro mult_mono' order_refl norm_X norm_ge_zero
```
```   225                 mult_nonneg_nonneg K)
```
```   226     also have "\<dots> = norm (Y n) * (B * K)"
```
```   227       by (simp only: mult_ac)
```
```   228     finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
```
```   229   qed
```
```   230 qed
```
```   231
```
```   232 lemma (in bounded_bilinear) Zseq_left:
```
```   233   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
```
```   234 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
```
```   235
```
```   236 lemma (in bounded_bilinear) Zseq_right:
```
```   237   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
```
```   238 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
```
```   239
```
```   240 lemmas Zseq_mult = mult.Zseq
```
```   241 lemmas Zseq_mult_right = mult.Zseq_right
```
```   242 lemmas Zseq_mult_left = mult.Zseq_left
```
```   243
```
```   244
```
```   245 subsection {* Limits of Sequences *}
```
```   246
```
```   247 lemma LIMSEQ_iff:
```
```   248   fixes L :: "'a::real_normed_vector"
```
```   249   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   250 unfolding LIMSEQ_def dist_norm ..
```
```   251
```
```   252 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
```
```   253 by (simp only: LIMSEQ_iff Zseq_def)
```
```   254
```
```   255 lemma metric_LIMSEQ_I:
```
```   256   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
```
```   257 by (simp add: LIMSEQ_def)
```
```   258
```
```   259 lemma metric_LIMSEQ_D:
```
```   260   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```   261 by (simp add: LIMSEQ_def)
```
```   262
```
```   263 lemma LIMSEQ_I:
```
```   264   fixes L :: "'a::real_normed_vector"
```
```   265   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   266 by (simp add: LIMSEQ_iff)
```
```   267
```
```   268 lemma LIMSEQ_D:
```
```   269   fixes L :: "'a::real_normed_vector"
```
```   270   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   271 by (simp add: LIMSEQ_iff)
```
```   272
```
```   273 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
```
```   274 by (simp add: LIMSEQ_def)
```
```   275
```
```   276 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
```
```   277 apply (safe intro!: LIMSEQ_const)
```
```   278 apply (rule ccontr)
```
```   279 apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
```
```   280 apply (simp add: zero_less_dist_iff)
```
```   281 apply auto
```
```   282 done
```
```   283
```
```   284 lemma LIMSEQ_norm:
```
```   285   fixes a :: "'a::real_normed_vector"
```
```   286   shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
```
```   287 unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)
```
```   288
```
```   289 lemma LIMSEQ_ignore_initial_segment:
```
```   290   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   291 apply (rule metric_LIMSEQ_I)
```
```   292 apply (drule (1) metric_LIMSEQ_D)
```
```   293 apply (erule exE, rename_tac N)
```
```   294 apply (rule_tac x=N in exI)
```
```   295 apply simp
```
```   296 done
```
```   297
```
```   298 lemma LIMSEQ_offset:
```
```   299   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   300 apply (rule metric_LIMSEQ_I)
```
```   301 apply (drule (1) metric_LIMSEQ_D)
```
```   302 apply (erule exE, rename_tac N)
```
```   303 apply (rule_tac x="N + k" in exI)
```
```   304 apply clarify
```
```   305 apply (drule_tac x="n - k" in spec)
```
```   306 apply (simp add: le_diff_conv2)
```
```   307 done
```
```   308
```
```   309 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   310 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   311
```
```   312 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   313 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   314
```
```   315 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   316 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   317
```
```   318 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   319   unfolding LIMSEQ_def
```
```   320   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   321
```
```   322 lemma LIMSEQ_add:
```
```   323   fixes a b :: "'a::real_normed_vector"
```
```   324   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
```
```   325 unfolding LIMSEQ_conv_tendsto by (rule tendsto_add)
```
```   326
```
```   327 lemma LIMSEQ_minus:
```
```   328   fixes a :: "'a::real_normed_vector"
```
```   329   shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
```
```   330 unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)
```
```   331
```
```   332 lemma LIMSEQ_minus_cancel:
```
```   333   fixes a :: "'a::real_normed_vector"
```
```   334   shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
```
```   335 by (drule LIMSEQ_minus, simp)
```
```   336
```
```   337 lemma LIMSEQ_diff:
```
```   338   fixes a b :: "'a::real_normed_vector"
```
```   339   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
```
```   340 unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)
```
```   341
```
```   342 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   343 apply (rule ccontr)
```
```   344 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   345 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   346 apply (clarify, rename_tac M N)
```
```   347 apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
```
```   348 apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
```
```   349 apply (erule le_less_trans, rule add_strict_mono, simp, simp)
```
```   350 apply (subst dist_commute, rule dist_triangle)
```
```   351 done
```
```   352
```
```   353 lemma (in bounded_linear) LIMSEQ:
```
```   354   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```
```   355 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   356
```
```   357 lemma (in bounded_bilinear) LIMSEQ:
```
```   358   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
```
```   359 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   360
```
```   361 lemma LIMSEQ_mult:
```
```   362   fixes a b :: "'a::real_normed_algebra"
```
```   363   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
```
```   364 by (rule mult.LIMSEQ)
```
```   365
```
```   366 lemma inverse_diff_inverse:
```
```   367   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   368    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   369 by (simp add: algebra_simps)
```
```   370
```
```   371 lemma Bseq_inverse_lemma:
```
```   372   fixes x :: "'a::real_normed_div_algebra"
```
```   373   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   374 apply (subst nonzero_norm_inverse, clarsimp)
```
```   375 apply (erule (1) le_imp_inverse_le)
```
```   376 done
```
```   377
```
```   378 lemma Bseq_inverse:
```
```   379   fixes a :: "'a::real_normed_div_algebra"
```
```   380   assumes X: "X ----> a"
```
```   381   assumes a: "a \<noteq> 0"
```
```   382   shows "Bseq (\<lambda>n. inverse (X n))"
```
```   383 proof -
```
```   384   from a have "0 < norm a" by simp
```
```   385   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   386   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   387   obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
```
```   388     using LIMSEQ_D [OF X r1] by fast
```
```   389   show ?thesis
```
```   390   proof (rule BseqI2' [rule_format])
```
```   391     fix n assume n: "N \<le> n"
```
```   392     hence 1: "norm (X n - a) < r" by (rule N)
```
```   393     hence 2: "X n \<noteq> 0" using r2 by auto
```
```   394     hence "norm (inverse (X n)) = inverse (norm (X n))"
```
```   395       by (rule nonzero_norm_inverse)
```
```   396     also have "\<dots> \<le> inverse (norm a - r)"
```
```   397     proof (rule le_imp_inverse_le)
```
```   398       show "0 < norm a - r" using r2 by simp
```
```   399     next
```
```   400       have "norm a - norm (X n) \<le> norm (a - X n)"
```
```   401         by (rule norm_triangle_ineq2)
```
```   402       also have "\<dots> = norm (X n - a)"
```
```   403         by (rule norm_minus_commute)
```
```   404       also have "\<dots> < r" using 1 .
```
```   405       finally show "norm a - r \<le> norm (X n)" by simp
```
```   406     qed
```
```   407     finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
```
```   408   qed
```
```   409 qed
```
```   410
```
```   411 lemma LIMSEQ_inverse_lemma:
```
```   412   fixes a :: "'a::real_normed_div_algebra"
```
```   413   shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
```
```   414          \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
```
```   415 apply (subst LIMSEQ_Zseq_iff)
```
```   416 apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
```
```   417 apply (rule Zseq_minus)
```
```   418 apply (rule Zseq_mult_left)
```
```   419 apply (rule mult.Bseq_prod_Zseq)
```
```   420 apply (erule (1) Bseq_inverse)
```
```   421 apply (simp add: LIMSEQ_Zseq_iff)
```
```   422 done
```
```   423
```
```   424 lemma LIMSEQ_inverse:
```
```   425   fixes a :: "'a::real_normed_div_algebra"
```
```   426   assumes X: "X ----> a"
```
```   427   assumes a: "a \<noteq> 0"
```
```   428   shows "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   429 proof -
```
```   430   from a have "0 < norm a" by simp
```
```   431   then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
```
```   432     using LIMSEQ_D [OF X] by fast
```
```   433   hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
```
```   434   hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
```
```   435
```
```   436   from X have "(\<lambda>n. X (n + k)) ----> a"
```
```   437     by (rule LIMSEQ_ignore_initial_segment)
```
```   438   hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
```
```   439     using a k by (rule LIMSEQ_inverse_lemma)
```
```   440   thus "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   441     by (rule LIMSEQ_offset)
```
```   442 qed
```
```   443
```
```   444 lemma LIMSEQ_divide:
```
```   445   fixes a b :: "'a::real_normed_field"
```
```   446   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
```
```   447 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
```
```   448
```
```   449 lemma LIMSEQ_pow:
```
```   450   fixes a :: "'a::{power, real_normed_algebra}"
```
```   451   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
```
```   452 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
```
```   453
```
```   454 lemma LIMSEQ_setsum:
```
```   455   fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   456   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   457   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
```
```   458 proof (cases "finite S")
```
```   459   case True
```
```   460   thus ?thesis using n
```
```   461   proof (induct)
```
```   462     case empty
```
```   463     show ?case
```
```   464       by (simp add: LIMSEQ_const)
```
```   465   next
```
```   466     case insert
```
```   467     thus ?case
```
```   468       by (simp add: LIMSEQ_add)
```
```   469   qed
```
```   470 next
```
```   471   case False
```
```   472   thus ?thesis
```
```   473     by (simp add: LIMSEQ_const)
```
```   474 qed
```
```   475
```
```   476 lemma LIMSEQ_setprod:
```
```   477   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
```
```   478   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   479   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
```
```   480 proof (cases "finite S")
```
```   481   case True
```
```   482   thus ?thesis using n
```
```   483   proof (induct)
```
```   484     case empty
```
```   485     show ?case
```
```   486       by (simp add: LIMSEQ_const)
```
```   487   next
```
```   488     case insert
```
```   489     thus ?case
```
```   490       by (simp add: LIMSEQ_mult)
```
```   491   qed
```
```   492 next
```
```   493   case False
```
```   494   thus ?thesis
```
```   495     by (simp add: setprod_def LIMSEQ_const)
```
```   496 qed
```
```   497
```
```   498 lemma LIMSEQ_add_const:
```
```   499   fixes a :: "'a::real_normed_vector"
```
```   500   shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
```
```   501 by (simp add: LIMSEQ_add LIMSEQ_const)
```
```   502
```
```   503 (* FIXME: delete *)
```
```   504 lemma LIMSEQ_add_minus:
```
```   505   fixes a b :: "'a::real_normed_vector"
```
```   506   shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
```
```   507 by (simp only: LIMSEQ_add LIMSEQ_minus)
```
```   508
```
```   509 lemma LIMSEQ_diff_const:
```
```   510   fixes a b :: "'a::real_normed_vector"
```
```   511   shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
```
```   512 by (simp add: LIMSEQ_diff LIMSEQ_const)
```
```   513
```
```   514 lemma LIMSEQ_diff_approach_zero:
```
```   515   fixes L :: "'a::real_normed_vector"
```
```   516   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```   517 by (drule (1) LIMSEQ_add, simp)
```
```   518
```
```   519 lemma LIMSEQ_diff_approach_zero2:
```
```   520   fixes L :: "'a::real_normed_vector"
```
```   521   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L";
```
```   522 by (drule (1) LIMSEQ_diff, simp)
```
```   523
```
```   524 text{*A sequence tends to zero iff its abs does*}
```
```   525 lemma LIMSEQ_norm_zero:
```
```   526   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   527   shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
```
```   528 by (simp add: LIMSEQ_iff)
```
```   529
```
```   530 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
```
```   531 by (simp add: LIMSEQ_iff)
```
```   532
```
```   533 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
```
```   534 by (drule LIMSEQ_norm, simp)
```
```   535
```
```   536 text{*An unbounded sequence's inverse tends to 0*}
```
```   537
```
```   538 lemma LIMSEQ_inverse_zero:
```
```   539   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   540 apply (rule LIMSEQ_I)
```
```   541 apply (drule_tac x="inverse r" in spec, safe)
```
```   542 apply (rule_tac x="N" in exI, safe)
```
```   543 apply (drule_tac x="n" in spec, safe)
```
```   544 apply (frule positive_imp_inverse_positive)
```
```   545 apply (frule (1) less_imp_inverse_less)
```
```   546 apply (subgoal_tac "0 < X n", simp)
```
```   547 apply (erule (1) order_less_trans)
```
```   548 done
```
```   549
```
```   550 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   551
```
```   552 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   553 apply (rule LIMSEQ_inverse_zero, safe)
```
```   554 apply (cut_tac x = r in reals_Archimedean2)
```
```   555 apply (safe, rule_tac x = n in exI)
```
```   556 apply (auto simp add: real_of_nat_Suc)
```
```   557 done
```
```   558
```
```   559 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   560 infinity is now easily proved*}
```
```   561
```
```   562 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   563      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   564 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   565
```
```   566 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   567      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   568 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   569
```
```   570 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   571      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   572 by (cut_tac b=1 in
```
```   573         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
```
```   574
```
```   575 lemma LIMSEQ_le_const:
```
```   576   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   577 apply (rule ccontr, simp only: linorder_not_le)
```
```   578 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
```
```   579 apply clarsimp
```
```   580 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
```
```   581 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
```
```   582 apply simp
```
```   583 done
```
```   584
```
```   585 lemma LIMSEQ_le_const2:
```
```   586   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   587 apply (subgoal_tac "- a \<le> - x", simp)
```
```   588 apply (rule LIMSEQ_le_const)
```
```   589 apply (erule LIMSEQ_minus)
```
```   590 apply simp
```
```   591 done
```
```   592
```
```   593 lemma LIMSEQ_le:
```
```   594   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
```
```   595 apply (subgoal_tac "0 \<le> y - x", simp)
```
```   596 apply (rule LIMSEQ_le_const)
```
```   597 apply (erule (1) LIMSEQ_diff)
```
```   598 apply (simp add: le_diff_eq)
```
```   599 done
```
```   600
```
```   601
```
```   602 subsection {* Convergence *}
```
```   603
```
```   604 lemma limI: "X ----> L ==> lim X = L"
```
```   605 apply (simp add: lim_def)
```
```   606 apply (blast intro: LIMSEQ_unique)
```
```   607 done
```
```   608
```
```   609 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   610 by (simp add: convergent_def)
```
```   611
```
```   612 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   613 by (auto simp add: convergent_def)
```
```   614
```
```   615 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   616 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   617
```
```   618 lemma convergent_minus_iff:
```
```   619   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   620   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```   621 apply (simp add: convergent_def)
```
```   622 apply (auto dest: LIMSEQ_minus)
```
```   623 apply (drule LIMSEQ_minus, auto)
```
```   624 done
```
```   625
```
```   626 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   627
```
```   628 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   629   unfolding Ex1_def
```
```   630   apply (rule_tac x="nat_rec e f" in exI)
```
```   631   apply (rule conjI)+
```
```   632 apply (rule def_nat_rec_0, simp)
```
```   633 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   634 apply (rule allI, rule impI, rule ext)
```
```   635 apply (erule conjE)
```
```   636 apply (induct_tac x)
```
```   637 apply (simp add: nat_rec_0)
```
```   638 apply (erule_tac x="n" in allE)
```
```   639 apply (simp)
```
```   640 done
```
```   641
```
```   642 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   643
```
```   644 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   645 apply (simp add: subseq_def)
```
```   646 apply (auto dest!: less_imp_Suc_add)
```
```   647 apply (induct_tac k)
```
```   648 apply (auto intro: less_trans)
```
```   649 done
```
```   650
```
```   651 lemma monoseq_Suc:
```
```   652    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
```
```   653                  | (\<forall>n. X (Suc n) \<le> X n))"
```
```   654 apply (simp add: monoseq_def)
```
```   655 apply (auto dest!: le_imp_less_or_eq)
```
```   656 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
```
```   657 apply (induct_tac "ka")
```
```   658 apply (auto intro: order_trans)
```
```   659 apply (erule contrapos_np)
```
```   660 apply (induct_tac "k")
```
```   661 apply (auto intro: order_trans)
```
```   662 done
```
```   663
```
```   664 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```   665 by (simp add: monoseq_def)
```
```   666
```
```   667 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```   668 by (simp add: monoseq_def)
```
```   669
```
```   670 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```   671 by (simp add: monoseq_Suc)
```
```   672
```
```   673 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```   674 by (simp add: monoseq_Suc)
```
```   675
```
```   676 lemma monoseq_minus: assumes "monoseq a"
```
```   677   shows "monoseq (\<lambda> n. - a n)"
```
```   678 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   679   case True
```
```   680   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   681   thus ?thesis by (rule monoI2)
```
```   682 next
```
```   683   case False
```
```   684   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   685   thus ?thesis by (rule monoI1)
```
```   686 qed
```
```   687
```
```   688 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
```
```   689   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
```
```   690          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   691 proof -
```
```   692   { fix x n fix a :: "nat \<Rightarrow> real"
```
```   693     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
```
```   694     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
```
```   695     have "a n \<le> x"
```
```   696     proof (rule ccontr)
```
```   697       assume "\<not> a n \<le> x" hence "x < a n" by auto
```
```   698       hence "0 < a n - x" by auto
```
```   699       from `a ----> x`[THEN LIMSEQ_D, OF this]
```
```   700       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
```
```   701       hence "norm (a (max no n) - x) < a n - x" by auto
```
```   702       moreover
```
```   703       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
```
```   704       hence "x < a (max no n)" by auto
```
```   705       ultimately
```
```   706       have "a (max no n) < a n" by auto
```
```   707       with monotone[where m=n and n="max no n"]
```
```   708       show False by auto
```
```   709     qed
```
```   710   } note top_down = this
```
```   711   { fix x n m fix a :: "nat \<Rightarrow> real"
```
```   712     assume "a ----> x" and "monoseq a" and "a m < x"
```
```   713     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
```
```   714     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   715       case True with top_down and `a ----> x` show ?thesis by auto
```
```   716     next
```
```   717       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
```
```   718       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
```
```   719       hence False using `a m < x` by auto
```
```   720       thus ?thesis ..
```
```   721     qed
```
```   722   } note when_decided = this
```
```   723
```
```   724   show ?thesis
```
```   725   proof (cases "\<exists> m. a m \<noteq> x")
```
```   726     case True then obtain m where "a m \<noteq> x" by auto
```
```   727     show ?thesis
```
```   728     proof (cases "a m < x")
```
```   729       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
```
```   730       show ?thesis by blast
```
```   731     next
```
```   732       case False hence "- a m < - x" using `a m \<noteq> x` by auto
```
```   733       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
```
```   734       show ?thesis by auto
```
```   735     qed
```
```   736   qed auto
```
```   737 qed
```
```   738
```
```   739 text{* for any sequence, there is a mootonic subsequence *}
```
```   740 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   741 proof-
```
```   742   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   743     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   744     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   745     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   746     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   747       using H apply -
```
```   748       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   749       unfolding order_le_less by blast
```
```   750     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   751     {fix n
```
```   752       have "?P (f (Suc n)) (f n)"
```
```   753 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   754 	using H apply -
```
```   755       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   756       unfolding order_le_less by blast
```
```   757     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   758   note fSuc = this
```
```   759     {fix p q assume pq: "p \<ge> f q"
```
```   760       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   761 	by (cases q, simp_all) }
```
```   762     note pqth = this
```
```   763     {fix q
```
```   764       have "f (Suc q) > f q" apply (induct q)
```
```   765 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   766     note fss = this
```
```   767     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   768     {fix a b
```
```   769       have "f a \<le> f (a + b)"
```
```   770       proof(induct b)
```
```   771 	case 0 thus ?case by simp
```
```   772       next
```
```   773 	case (Suc b)
```
```   774 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   775       qed}
```
```   776     note fmon0 = this
```
```   777     have "monoseq (\<lambda>n. s (f n))"
```
```   778     proof-
```
```   779       {fix n
```
```   780 	have "s (f n) \<ge> s (f (Suc n))"
```
```   781 	proof(cases n)
```
```   782 	  case 0
```
```   783 	  assume n0: "n = 0"
```
```   784 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   785 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   786 	next
```
```   787 	  case (Suc m)
```
```   788 	  assume m: "n = Suc m"
```
```   789 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   790 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   791 	qed}
```
```   792       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   793     qed
```
```   794     with th1 have ?thesis by blast}
```
```   795   moreover
```
```   796   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   797     {fix p assume p: "p \<ge> Suc N"
```
```   798       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   799       have "m \<noteq> p" using m(2) by auto
```
```   800       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   801     note th0 = this
```
```   802     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   803     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   804     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   805       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   806     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   807       using N apply -
```
```   808       apply (erule allE[where x="Suc N"], clarsimp)
```
```   809       apply (rule_tac x="m" in exI)
```
```   810       apply auto
```
```   811       apply (subgoal_tac "Suc N \<noteq> m")
```
```   812       apply simp
```
```   813       apply (rule ccontr, simp)
```
```   814       done
```
```   815     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   816     {fix n
```
```   817       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   818 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   819       proof (induct n)
```
```   820 	case 0 thus ?case
```
```   821 	  using f0 N apply auto
```
```   822 	  apply (erule allE[where x="f 0"], clarsimp)
```
```   823 	  apply (rule_tac x="m" in exI, simp)
```
```   824 	  by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   825       next
```
```   826 	case (Suc n)
```
```   827 	from Suc.hyps have Nfn: "N < f n" by blast
```
```   828 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   829 	with Nfn have mN: "m > N" by arith
```
```   830 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   831
```
```   832 	from key have th0: "f (Suc n) > N" by simp
```
```   833 	from N[rule_format, OF th0]
```
```   834 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   835 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   836 	hence "m' > f (Suc n)" using m'(1) by simp
```
```   837 	with key m'(2) show ?case by auto
```
```   838       qed}
```
```   839     note fSuc = this
```
```   840     {fix n
```
```   841       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
```
```   842       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   843     note thf = this
```
```   844     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   845     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   846       apply -
```
```   847       apply (rule disjI1)
```
```   848       apply auto
```
```   849       apply (rule order_less_imp_le)
```
```   850       apply blast
```
```   851       done
```
```   852     then have ?thesis  using sqf by blast}
```
```   853   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   854 qed
```
```   855
```
```   856 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   857 proof(induct n)
```
```   858   case 0 thus ?case by simp
```
```   859 next
```
```   860   case (Suc n)
```
```   861   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   862   have "n < f (Suc n)" by arith
```
```   863   thus ?case by arith
```
```   864 qed
```
```   865
```
```   866 lemma LIMSEQ_subseq_LIMSEQ:
```
```   867   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```   868 apply (auto simp add: LIMSEQ_def)
```
```   869 apply (drule_tac x=r in spec, clarify)
```
```   870 apply (rule_tac x=no in exI, clarify)
```
```   871 apply (blast intro: seq_suble le_trans dest!: spec)
```
```   872 done
```
```   873
```
```   874 subsection {* Bounded Monotonic Sequences *}
```
```   875
```
```   876
```
```   877 text{*Bounded Sequence*}
```
```   878
```
```   879 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   880 by (simp add: Bseq_def)
```
```   881
```
```   882 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   883 by (auto simp add: Bseq_def)
```
```   884
```
```   885 lemma lemma_NBseq_def:
```
```   886      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
```
```   887       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   888 apply auto
```
```   889  prefer 2 apply force
```
```   890 apply (cut_tac x = K in reals_Archimedean2, clarify)
```
```   891 apply (rule_tac x = n in exI, clarify)
```
```   892 apply (drule_tac x = na in spec)
```
```   893 apply (auto simp add: real_of_nat_Suc)
```
```   894 done
```
```   895
```
```   896 text{* alternative definition for Bseq *}
```
```   897 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   898 apply (simp add: Bseq_def)
```
```   899 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   900 done
```
```   901
```
```   902 lemma lemma_NBseq_def2:
```
```   903      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   904 apply (subst lemma_NBseq_def, auto)
```
```   905 apply (rule_tac x = "Suc N" in exI)
```
```   906 apply (rule_tac [2] x = N in exI)
```
```   907 apply (auto simp add: real_of_nat_Suc)
```
```   908  prefer 2 apply (blast intro: order_less_imp_le)
```
```   909 apply (drule_tac x = n in spec, simp)
```
```   910 done
```
```   911
```
```   912 (* yet another definition for Bseq *)
```
```   913 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   914 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   915
```
```   916 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   917
```
```   918 lemma Bseq_isUb:
```
```   919   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   920 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   921
```
```   922
```
```   923 text{* Use completeness of reals (supremum property)
```
```   924    to show that any bounded sequence has a least upper bound*}
```
```   925
```
```   926 lemma Bseq_isLub:
```
```   927   "!!(X::nat=>real). Bseq X ==>
```
```   928    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   929 by (blast intro: reals_complete Bseq_isUb)
```
```   930
```
```   931 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   932
```
```   933 lemma lemma_converg1:
```
```   934      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
```
```   935                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
```
```   936                |] ==> \<forall>n \<ge> ma. X n = X ma"
```
```   937 apply safe
```
```   938 apply (drule_tac y = "X n" in isLubD2)
```
```   939 apply (blast dest: order_antisym)+
```
```   940 done
```
```   941
```
```   942 text{* The best of both worlds: Easier to prove this result as a standard
```
```   943    theorem and then use equivalence to "transfer" it into the
```
```   944    equivalent nonstandard form if needed!*}
```
```   945
```
```   946 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   947 apply (simp add: LIMSEQ_def)
```
```   948 apply (rule_tac x = "X m" in exI, safe)
```
```   949 apply (rule_tac x = m in exI, safe)
```
```   950 apply (drule spec, erule impE, auto)
```
```   951 done
```
```   952
```
```   953 lemma lemma_converg2:
```
```   954    "!!(X::nat=>real).
```
```   955     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
```
```   956 apply safe
```
```   957 apply (drule_tac y = "X m" in isLubD2)
```
```   958 apply (auto dest!: order_le_imp_less_or_eq)
```
```   959 done
```
```   960
```
```   961 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
```
```   962 by (rule setleI [THEN isUbI], auto)
```
```   963
```
```   964 text{* FIXME: @{term "U - T < U"} is redundant *}
```
```   965 lemma lemma_converg4: "!!(X::nat=> real).
```
```   966                [| \<forall>m. X m ~= U;
```
```   967                   isLub UNIV {x. \<exists>n. X n = x} U;
```
```   968                   0 < T;
```
```   969                   U + - T < U
```
```   970                |] ==> \<exists>m. U + -T < X m & X m < U"
```
```   971 apply (drule lemma_converg2, assumption)
```
```   972 apply (rule ccontr, simp)
```
```   973 apply (simp add: linorder_not_less)
```
```   974 apply (drule lemma_converg3)
```
```   975 apply (drule isLub_le_isUb, assumption)
```
```   976 apply (auto dest: order_less_le_trans)
```
```   977 done
```
```   978
```
```   979 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```   980
```
```   981 lemma Bseq_mono_convergent:
```
```   982      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
```
```   983 apply (simp add: convergent_def)
```
```   984 apply (frule Bseq_isLub, safe)
```
```   985 apply (case_tac "\<exists>m. X m = U", auto)
```
```   986 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
```
```   987 (* second case *)
```
```   988 apply (rule_tac x = U in exI)
```
```   989 apply (subst LIMSEQ_iff, safe)
```
```   990 apply (frule lemma_converg2, assumption)
```
```   991 apply (drule lemma_converg4, auto)
```
```   992 apply (rule_tac x = m in exI, safe)
```
```   993 apply (subgoal_tac "X m \<le> X n")
```
```   994  prefer 2 apply blast
```
```   995 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
```
```   996 done
```
```   997
```
```   998 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```   999 by (simp add: Bseq_def)
```
```  1000
```
```  1001 text{*Main monotonicity theorem*}
```
```  1002 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
```
```  1003 apply (simp add: monoseq_def, safe)
```
```  1004 apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
```
```  1005 apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
```
```  1006 apply (auto intro!: Bseq_mono_convergent)
```
```  1007 done
```
```  1008
```
```  1009 subsubsection{*Increasing and Decreasing Series*}
```
```  1010
```
```  1011 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```  1012   by (simp add: incseq_def monoseq_def)
```
```  1013
```
```  1014 lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
```
```  1015   using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
```
```  1016 proof
```
```  1017   assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
```
```  1018   thus ?thesis by simp
```
```  1019 next
```
```  1020   assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
```
```  1021   hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
```
```  1022     by (auto simp add: incseq_def intro: order_antisym)
```
```  1023   have X: "!!n. X n = X 0"
```
```  1024     by (blast intro: const [of 0])
```
```  1025   have "X = (\<lambda>n. X 0)"
```
```  1026     by (blast intro: ext X)
```
```  1027   hence "L = X 0" using LIMSEQ_const [of "X 0"]
```
```  1028     by (auto intro: LIMSEQ_unique lim)
```
```  1029   thus ?thesis
```
```  1030     by (blast intro: eq_refl X)
```
```  1031 qed
```
```  1032
```
```  1033 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```  1034   by (simp add: decseq_def monoseq_def)
```
```  1035
```
```  1036 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
```
```  1037   by (simp add: decseq_def incseq_def)
```
```  1038
```
```  1039
```
```  1040 lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
```
```  1041 proof -
```
```  1042   have inc: "incseq (\<lambda>n. - X n)" using dec
```
```  1043     by (simp add: decseq_eq_incseq)
```
```  1044   have "- X n \<le> - L"
```
```  1045     by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim)
```
```  1046   thus ?thesis
```
```  1047     by simp
```
```  1048 qed
```
```  1049
```
```  1050 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```  1051
```
```  1052 text{*alternative formulation for boundedness*}
```
```  1053 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```  1054 apply (unfold Bseq_def, safe)
```
```  1055 apply (rule_tac [2] x = "k + norm x" in exI)
```
```  1056 apply (rule_tac x = K in exI, simp)
```
```  1057 apply (rule exI [where x = 0], auto)
```
```  1058 apply (erule order_less_le_trans, simp)
```
```  1059 apply (drule_tac x=n in spec, fold diff_def)
```
```  1060 apply (drule order_trans [OF norm_triangle_ineq2])
```
```  1061 apply simp
```
```  1062 done
```
```  1063
```
```  1064 text{*alternative formulation for boundedness*}
```
```  1065 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```  1066 apply safe
```
```  1067 apply (simp add: Bseq_def, safe)
```
```  1068 apply (rule_tac x = "K + norm (X N)" in exI)
```
```  1069 apply auto
```
```  1070 apply (erule order_less_le_trans, simp)
```
```  1071 apply (rule_tac x = N in exI, safe)
```
```  1072 apply (drule_tac x = n in spec)
```
```  1073 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```  1074 apply (auto simp add: Bseq_iff2)
```
```  1075 done
```
```  1076
```
```  1077 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```  1078 apply (simp add: Bseq_def)
```
```  1079 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```  1080 apply (drule_tac x = n in spec, arith)
```
```  1081 done
```
```  1082
```
```  1083
```
```  1084 subsection {* Cauchy Sequences *}
```
```  1085
```
```  1086 lemma metric_CauchyI:
```
```  1087   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1088 by (simp add: Cauchy_def)
```
```  1089
```
```  1090 lemma metric_CauchyD:
```
```  1091   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```  1092 by (simp add: Cauchy_def)
```
```  1093
```
```  1094 lemma Cauchy_iff:
```
```  1095   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1096   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1097 unfolding Cauchy_def dist_norm ..
```
```  1098
```
```  1099 lemma CauchyI:
```
```  1100   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1101   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"
```
```  1102 by (simp add: Cauchy_iff)
```
```  1103
```
```  1104 lemma CauchyD:
```
```  1105   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1106   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1107 by (simp add: Cauchy_iff)
```
```  1108
```
```  1109 lemma Cauchy_subseq_Cauchy:
```
```  1110   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```  1111 apply (auto simp add: Cauchy_def)
```
```  1112 apply (drule_tac x=e in spec, clarify)
```
```  1113 apply (rule_tac x=M in exI, clarify)
```
```  1114 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```  1115 done
```
```  1116
```
```  1117 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1118
```
```  1119 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1120   proof mechanization rather than the nonstandard proof*}
```
```  1121
```
```  1122 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1123           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1124 apply (clarify, drule spec, drule (1) mp)
```
```  1125 apply (simp only: norm_minus_commute)
```
```  1126 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1127 apply simp
```
```  1128 done
```
```  1129
```
```  1130 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```  1131 apply (simp add: Cauchy_iff)
```
```  1132 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```  1133 apply (drule_tac x="M" in spec, simp)
```
```  1134 apply (drule lemmaCauchy)
```
```  1135 apply (rule_tac k="M" in Bseq_offset)
```
```  1136 apply (simp add: Bseq_def)
```
```  1137 apply (rule_tac x="1 + norm (X M)" in exI)
```
```  1138 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```  1139 apply (simp add: order_less_imp_le)
```
```  1140 done
```
```  1141
```
```  1142 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1143
```
```  1144 axclass banach \<subseteq> real_normed_vector
```
```  1145   Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1146
```
```  1147 theorem LIMSEQ_imp_Cauchy:
```
```  1148   assumes X: "X ----> a" shows "Cauchy X"
```
```  1149 proof (rule metric_CauchyI)
```
```  1150   fix e::real assume "0 < e"
```
```  1151   hence "0 < e/2" by simp
```
```  1152   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```  1153   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```  1154   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```  1155   proof (intro exI allI impI)
```
```  1156     fix m assume "N \<le> m"
```
```  1157     hence m: "dist (X m) a < e/2" using N by fast
```
```  1158     fix n assume "N \<le> n"
```
```  1159     hence n: "dist (X n) a < e/2" using N by fast
```
```  1160     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```  1161       by (rule dist_triangle2)
```
```  1162     also from m n have "\<dots> < e" by simp
```
```  1163     finally show "dist (X m) (X n) < e" .
```
```  1164   qed
```
```  1165 qed
```
```  1166
```
```  1167 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1168 unfolding convergent_def
```
```  1169 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1170
```
```  1171 text {*
```
```  1172 Proof that Cauchy sequences converge based on the one from
```
```  1173 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1174 *}
```
```  1175
```
```  1176 text {*
```
```  1177   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1178   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1179 *}
```
```  1180
```
```  1181 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1182 by (simp add: isUbI setleI)
```
```  1183
```
```  1184 lemma real_abs_diff_less_iff:
```
```  1185   "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
```
```  1186 by auto
```
```  1187
```
```  1188 locale real_Cauchy =
```
```  1189   fixes X :: "nat \<Rightarrow> real"
```
```  1190   assumes X: "Cauchy X"
```
```  1191   fixes S :: "real set"
```
```  1192   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1193
```
```  1194 lemma real_CauchyI:
```
```  1195   assumes "Cauchy X"
```
```  1196   shows "real_Cauchy X"
```
```  1197   proof qed (fact assms)
```
```  1198
```
```  1199 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
```
```  1200 by (unfold S_def, auto)
```
```  1201
```
```  1202 lemma (in real_Cauchy) bound_isUb:
```
```  1203   assumes N: "\<forall>n\<ge>N. X n < x"
```
```  1204   shows "isUb UNIV S x"
```
```  1205 proof (rule isUb_UNIV_I)
```
```  1206   fix y::real assume "y \<in> S"
```
```  1207   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1208     by (simp add: S_def)
```
```  1209   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1210   hence "y < X (max M N)" by simp
```
```  1211   also have "\<dots> < x" using N by simp
```
```  1212   finally show "y \<le> x"
```
```  1213     by (rule order_less_imp_le)
```
```  1214 qed
```
```  1215
```
```  1216 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
```
```  1217 proof (rule reals_complete)
```
```  1218   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```  1219     using CauchyD [OF X zero_less_one] by fast
```
```  1220   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```  1221   show "\<exists>x. x \<in> S"
```
```  1222   proof
```
```  1223     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1224       by (simp add: real_abs_diff_less_iff)
```
```  1225     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1226   qed
```
```  1227   show "\<exists>u. isUb UNIV S u"
```
```  1228   proof
```
```  1229     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1230       by (simp add: real_abs_diff_less_iff)
```
```  1231     thus "isUb UNIV S (X N + 1)"
```
```  1232       by (rule bound_isUb)
```
```  1233   qed
```
```  1234 qed
```
```  1235
```
```  1236 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
```
```  1237   assumes x: "isLub UNIV S x"
```
```  1238   shows "X ----> x"
```
```  1239 proof (rule LIMSEQ_I)
```
```  1240   fix r::real assume "0 < r"
```
```  1241   hence r: "0 < r/2" by simp
```
```  1242   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```  1243     using CauchyD [OF X r] by fast
```
```  1244   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```  1245   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1246     by (simp only: real_norm_def real_abs_diff_less_iff)
```
```  1247
```
```  1248   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1249   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1250   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1251
```
```  1252   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1253   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1254   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1255
```
```  1256   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```  1257   proof (intro exI allI impI)
```
```  1258     fix n assume n: "N \<le> n"
```
```  1259     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1260     thus "norm (X n - x) < r" using 1 2
```
```  1261       by (simp add: real_abs_diff_less_iff)
```
```  1262   qed
```
```  1263 qed
```
```  1264
```
```  1265 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
```
```  1266 proof -
```
```  1267   obtain x where "isLub UNIV S x"
```
```  1268     using isLub_ex by fast
```
```  1269   hence "X ----> x"
```
```  1270     by (rule isLub_imp_LIMSEQ)
```
```  1271   thus ?thesis ..
```
```  1272 qed
```
```  1273
```
```  1274 lemma real_Cauchy_convergent:
```
```  1275   fixes X :: "nat \<Rightarrow> real"
```
```  1276   shows "Cauchy X \<Longrightarrow> convergent X"
```
```  1277 unfolding convergent_def
```
```  1278 by (rule real_Cauchy.LIMSEQ_ex)
```
```  1279  (rule real_CauchyI)
```
```  1280
```
```  1281 instance real :: banach
```
```  1282 by intro_classes (rule real_Cauchy_convergent)
```
```  1283
```
```  1284 lemma Cauchy_convergent_iff:
```
```  1285   fixes X :: "nat \<Rightarrow> 'a::banach"
```
```  1286   shows "Cauchy X = convergent X"
```
```  1287 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1288
```
```  1289 lemma convergent_subseq_convergent:
```
```  1290   fixes X :: "nat \<Rightarrow> 'a::banach"
```
```  1291   shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```  1292   by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
```
```  1293
```
```  1294
```
```  1295 subsection {* Power Sequences *}
```
```  1296
```
```  1297 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1298 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1299   also fact that bounded and monotonic sequence converges.*}
```
```  1300
```
```  1301 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1302 apply (simp add: Bseq_def)
```
```  1303 apply (rule_tac x = 1 in exI)
```
```  1304 apply (simp add: power_abs)
```
```  1305 apply (auto dest: power_mono)
```
```  1306 done
```
```  1307
```
```  1308 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1309 apply (clarify intro!: mono_SucI2)
```
```  1310 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1311 done
```
```  1312
```
```  1313 lemma convergent_realpow:
```
```  1314   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1315 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1316
```
```  1317 lemma LIMSEQ_inverse_realpow_zero_lemma:
```
```  1318   fixes x :: real
```
```  1319   assumes x: "0 \<le> x"
```
```  1320   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1321 apply (induct n)
```
```  1322 apply simp
```
```  1323 apply simp
```
```  1324 apply (rule order_trans)
```
```  1325 prefer 2
```
```  1326 apply (erule mult_left_mono)
```
```  1327 apply (rule add_increasing [OF x], simp)
```
```  1328 apply (simp add: real_of_nat_Suc)
```
```  1329 apply (simp add: ring_distribs)
```
```  1330 apply (simp add: mult_nonneg_nonneg x)
```
```  1331 done
```
```  1332
```
```  1333 lemma LIMSEQ_inverse_realpow_zero:
```
```  1334   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1335 proof (rule LIMSEQ_inverse_zero [rule_format])
```
```  1336   fix y :: real
```
```  1337   assume x: "1 < x"
```
```  1338   hence "0 < x - 1" by simp
```
```  1339   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
```
```  1340     by (rule reals_Archimedean3)
```
```  1341   hence "\<exists>N::nat. y < real N * (x - 1)" ..
```
```  1342   then obtain N::nat where "y < real N * (x - 1)" ..
```
```  1343   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
```
```  1344   also have "\<dots> \<le> (x - 1 + 1) ^ N"
```
```  1345     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
```
```  1346   also have "\<dots> = x ^ N" by simp
```
```  1347   finally have "y < x ^ N" .
```
```  1348   hence "\<forall>n\<ge>N. y < x ^ n"
```
```  1349     apply clarify
```
```  1350     apply (erule order_less_le_trans)
```
```  1351     apply (erule power_increasing)
```
```  1352     apply (rule order_less_imp_le [OF x])
```
```  1353     done
```
```  1354   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
```
```  1355 qed
```
```  1356
```
```  1357 lemma LIMSEQ_realpow_zero:
```
```  1358   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1359 proof (cases)
```
```  1360   assume "x = 0"
```
```  1361   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
```
```  1362   thus ?thesis by (rule LIMSEQ_imp_Suc)
```
```  1363 next
```
```  1364   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1365   hence x0: "0 < x" by simp
```
```  1366   assume x1: "x < 1"
```
```  1367   from x0 x1 have "1 < inverse x"
```
```  1368     by (rule real_inverse_gt_one)
```
```  1369   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1370     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1371   thus ?thesis by (simp add: power_inverse)
```
```  1372 qed
```
```  1373
```
```  1374 lemma LIMSEQ_power_zero:
```
```  1375   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1376   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1377 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1378 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
```
```  1379 apply (simp add: power_abs norm_power_ineq)
```
```  1380 done
```
```  1381
```
```  1382 lemma LIMSEQ_divide_realpow_zero:
```
```  1383   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
```
```  1384 apply (cut_tac a = a and x1 = "inverse x" in
```
```  1385         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
```
```  1386 apply (auto simp add: divide_inverse power_inverse)
```
```  1387 apply (simp add: inverse_eq_divide pos_divide_less_eq)
```
```  1388 done
```
```  1389
```
```  1390 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1391
```
```  1392 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
```
```  1393 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1394
```
```  1395 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
```
```  1396 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
```
```  1397 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
```
```  1398 done
```
```  1399
```
```  1400 end
```