src/HOL/SEQ.thy
 author huffman Fri Jun 12 15:46:21 2009 -0700 (2009-06-12) changeset 31588 2651f172c38b parent 31488 5691ccb8d6b5 child 32064 53ca12ff305d permissions -rw-r--r--
```     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   Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
```
```    17     --{*Standard definition of sequence converging to zero*}
```
```    18   [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
```
```    19
```
```    20 definition
```
```    21   LIMSEQ :: "[nat \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
```
```    22     ("((_)/ ----> (_))" [60, 60] 60) where
```
```    23     --{*Standard definition of convergence of sequence*}
```
```    24   [code del]: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```    25
```
```    26 definition
```
```    27   lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
```
```    28     --{*Standard definition of limit using choice operator*}
```
```    29   "lim X = (THE L. X ----> L)"
```
```    30
```
```    31 definition
```
```    32   convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    33     --{*Standard definition of convergence*}
```
```    34   "convergent X = (\<exists>L. X ----> L)"
```
```    35
```
```    36 definition
```
```    37   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
```
```    38     --{*Standard definition for bounded sequence*}
```
```    39   [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
```
```    40
```
```    41 definition
```
```    42   monoseq :: "(nat=>real)=>bool" where
```
```    43     --{*Definition of monotonicity.
```
```    44         The use of disjunction here complicates proofs considerably.
```
```    45         One alternative is to add a Boolean argument to indicate the direction.
```
```    46         Another is to develop the notions of increasing and decreasing first.*}
```
```    47   [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))"
```
```    48
```
```    49 definition
```
```    50   incseq :: "(nat=>real)=>bool" where
```
```    51     --{*Increasing sequence*}
```
```    52   [code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
```
```    53
```
```    54 definition
```
```    55   decseq :: "(nat=>real)=>bool" where
```
```    56     --{*Increasing sequence*}
```
```    57   [code del]: "decseq X = (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```    58
```
```    59 definition
```
```    60   subseq :: "(nat => nat) => bool" where
```
```    61     --{*Definition of subsequence*}
```
```    62   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
```
```    63
```
```    64 definition
```
```    65   Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    66     --{*Standard definition of the Cauchy condition*}
```
```    67   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```    68
```
```    69
```
```    70 subsection {* Bounded Sequences *}
```
```    71
```
```    72 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
```
```    73 unfolding Bseq_def
```
```    74 proof (intro exI conjI allI)
```
```    75   show "0 < max K 1" by simp
```
```    76 next
```
```    77   fix n::nat
```
```    78   have "norm (X n) \<le> K" by (rule K)
```
```    79   thus "norm (X n) \<le> max K 1" by simp
```
```    80 qed
```
```    81
```
```    82 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```    83 unfolding Bseq_def by auto
```
```    84
```
```    85 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
```
```    86 proof (rule BseqI')
```
```    87   let ?A = "norm ` X ` {..N}"
```
```    88   have 1: "finite ?A" by simp
```
```    89   fix n::nat
```
```    90   show "norm (X n) \<le> max K (Max ?A)"
```
```    91   proof (cases rule: linorder_le_cases)
```
```    92     assume "n \<ge> N"
```
```    93     hence "norm (X n) \<le> K" using K by simp
```
```    94     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```    95   next
```
```    96     assume "n \<le> N"
```
```    97     hence "norm (X n) \<in> ?A" by simp
```
```    98     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
```
```    99     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   100   qed
```
```   101 qed
```
```   102
```
```   103 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```   104 unfolding Bseq_def by auto
```
```   105
```
```   106 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```   107 apply (erule BseqE)
```
```   108 apply (rule_tac N="k" and K="K" in BseqI2')
```
```   109 apply clarify
```
```   110 apply (drule_tac x="n - k" in spec, simp)
```
```   111 done
```
```   112
```
```   113 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
```
```   114 unfolding Bfun_def eventually_sequentially
```
```   115 apply (rule iffI)
```
```   116 apply (simp add: Bseq_def, fast)
```
```   117 apply (fast intro: BseqI2')
```
```   118 done
```
```   119
```
```   120
```
```   121 subsection {* Sequences That Converge to Zero *}
```
```   122
```
```   123 lemma ZseqI:
```
```   124   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
```
```   125 unfolding Zseq_def by simp
```
```   126
```
```   127 lemma ZseqD:
```
```   128   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
```
```   129 unfolding Zseq_def by simp
```
```   130
```
```   131 lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
```
```   132 unfolding Zseq_def Zfun_def eventually_sequentially ..
```
```   133
```
```   134 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
```
```   135 unfolding Zseq_def by simp
```
```   136
```
```   137 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
```
```   138 unfolding Zseq_def by force
```
```   139
```
```   140 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
```
```   141 unfolding Zseq_def by simp
```
```   142
```
```   143 lemma Zseq_imp_Zseq:
```
```   144   assumes X: "Zseq X"
```
```   145   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
```
```   146   shows "Zseq (\<lambda>n. Y n)"
```
```   147 using X Y Zfun_imp_Zfun [of X sequentially Y K]
```
```   148 unfolding Zseq_conv_Zfun by simp
```
```   149
```
```   150 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
```
```   151 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
```
```   152
```
```   153 lemma Zseq_add:
```
```   154   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"
```
```   155 unfolding Zseq_conv_Zfun by (rule Zfun_add)
```
```   156
```
```   157 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
```
```   158 unfolding Zseq_def by simp
```
```   159
```
```   160 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
```
```   161 by (simp only: diff_minus Zseq_add Zseq_minus)
```
```   162
```
```   163 lemma (in bounded_linear) Zseq:
```
```   164   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
```
```   165 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   166
```
```   167 lemma (in bounded_bilinear) Zseq:
```
```   168   "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   169 unfolding Zseq_conv_Zfun by (rule Zfun)
```
```   170
```
```   171 lemma (in bounded_bilinear) Zseq_prod_Bseq:
```
```   172   "Zseq X \<Longrightarrow> Bseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   173 unfolding Zseq_conv_Zfun Bseq_conv_Bfun
```
```   174 by (rule Zfun_prod_Bfun)
```
```   175
```
```   176 lemma (in bounded_bilinear) Bseq_prod_Zseq:
```
```   177   "Bseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
```
```   178 unfolding Zseq_conv_Zfun Bseq_conv_Bfun
```
```   179 by (rule Bfun_prod_Zfun)
```
```   180
```
```   181 lemma (in bounded_bilinear) Zseq_left:
```
```   182   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
```
```   183 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
```
```   184
```
```   185 lemma (in bounded_bilinear) Zseq_right:
```
```   186   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
```
```   187 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
```
```   188
```
```   189 lemmas Zseq_mult = mult.Zseq
```
```   190 lemmas Zseq_mult_right = mult.Zseq_right
```
```   191 lemmas Zseq_mult_left = mult.Zseq_left
```
```   192
```
```   193
```
```   194 subsection {* Limits of Sequences *}
```
```   195
```
```   196 lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> (X ---> L) sequentially"
```
```   197 unfolding LIMSEQ_def tendsto_iff eventually_sequentially ..
```
```   198
```
```   199 lemma LIMSEQ_iff:
```
```   200   fixes L :: "'a::real_normed_vector"
```
```   201   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   202 unfolding LIMSEQ_def dist_norm ..
```
```   203
```
```   204 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
```
```   205 by (simp only: LIMSEQ_iff Zseq_def)
```
```   206
```
```   207 lemma metric_LIMSEQ_I:
```
```   208   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
```
```   209 by (simp add: LIMSEQ_def)
```
```   210
```
```   211 lemma metric_LIMSEQ_D:
```
```   212   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```   213 by (simp add: LIMSEQ_def)
```
```   214
```
```   215 lemma LIMSEQ_I:
```
```   216   fixes L :: "'a::real_normed_vector"
```
```   217   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   218 by (simp add: LIMSEQ_iff)
```
```   219
```
```   220 lemma LIMSEQ_D:
```
```   221   fixes L :: "'a::real_normed_vector"
```
```   222   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   223 by (simp add: LIMSEQ_iff)
```
```   224
```
```   225 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
```
```   226 by (simp add: LIMSEQ_def)
```
```   227
```
```   228 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
```
```   229 apply (safe intro!: LIMSEQ_const)
```
```   230 apply (rule ccontr)
```
```   231 apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
```
```   232 apply (simp add: zero_less_dist_iff)
```
```   233 apply auto
```
```   234 done
```
```   235
```
```   236 lemma LIMSEQ_norm:
```
```   237   fixes a :: "'a::real_normed_vector"
```
```   238   shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
```
```   239 unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)
```
```   240
```
```   241 lemma LIMSEQ_ignore_initial_segment:
```
```   242   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   243 apply (rule metric_LIMSEQ_I)
```
```   244 apply (drule (1) metric_LIMSEQ_D)
```
```   245 apply (erule exE, rename_tac N)
```
```   246 apply (rule_tac x=N in exI)
```
```   247 apply simp
```
```   248 done
```
```   249
```
```   250 lemma LIMSEQ_offset:
```
```   251   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   252 apply (rule metric_LIMSEQ_I)
```
```   253 apply (drule (1) metric_LIMSEQ_D)
```
```   254 apply (erule exE, rename_tac N)
```
```   255 apply (rule_tac x="N + k" in exI)
```
```   256 apply clarify
```
```   257 apply (drule_tac x="n - k" in spec)
```
```   258 apply (simp add: le_diff_conv2)
```
```   259 done
```
```   260
```
```   261 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   262 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   263
```
```   264 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   265 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   266
```
```   267 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   268 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   269
```
```   270 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   271   unfolding LIMSEQ_def
```
```   272   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   273
```
```   274 lemma LIMSEQ_add:
```
```   275   fixes a b :: "'a::real_normed_vector"
```
```   276   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
```
```   277 unfolding LIMSEQ_conv_tendsto by (rule tendsto_add)
```
```   278
```
```   279 lemma LIMSEQ_minus:
```
```   280   fixes a :: "'a::real_normed_vector"
```
```   281   shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
```
```   282 unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)
```
```   283
```
```   284 lemma LIMSEQ_minus_cancel:
```
```   285   fixes a :: "'a::real_normed_vector"
```
```   286   shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
```
```   287 by (drule LIMSEQ_minus, simp)
```
```   288
```
```   289 lemma LIMSEQ_diff:
```
```   290   fixes a b :: "'a::real_normed_vector"
```
```   291   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
```
```   292 unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)
```
```   293
```
```   294 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   295 apply (rule ccontr)
```
```   296 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   297 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
```
```   298 apply (clarify, rename_tac M N)
```
```   299 apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
```
```   300 apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
```
```   301 apply (erule le_less_trans, rule add_strict_mono, simp, simp)
```
```   302 apply (subst dist_commute, rule dist_triangle)
```
```   303 done
```
```   304
```
```   305 lemma (in bounded_linear) LIMSEQ:
```
```   306   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```
```   307 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   308
```
```   309 lemma (in bounded_bilinear) LIMSEQ:
```
```   310   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
```
```   311 unfolding LIMSEQ_conv_tendsto by (rule tendsto)
```
```   312
```
```   313 lemma LIMSEQ_mult:
```
```   314   fixes a b :: "'a::real_normed_algebra"
```
```   315   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
```
```   316 by (rule mult.LIMSEQ)
```
```   317
```
```   318 lemma Bseq_inverse_lemma:
```
```   319   fixes x :: "'a::real_normed_div_algebra"
```
```   320   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   321 apply (subst nonzero_norm_inverse, clarsimp)
```
```   322 apply (erule (1) le_imp_inverse_le)
```
```   323 done
```
```   324
```
```   325 lemma Bseq_inverse:
```
```   326   fixes a :: "'a::real_normed_div_algebra"
```
```   327   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```   328 unfolding LIMSEQ_conv_tendsto Bseq_conv_Bfun
```
```   329 by (rule Bfun_inverse)
```
```   330
```
```   331 lemma LIMSEQ_inverse:
```
```   332   fixes a :: "'a::real_normed_div_algebra"
```
```   333   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
```
```   334 unfolding LIMSEQ_conv_tendsto
```
```   335 by (rule tendsto_inverse)
```
```   336
```
```   337 lemma LIMSEQ_divide:
```
```   338   fixes a b :: "'a::real_normed_field"
```
```   339   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
```
```   340 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
```
```   341
```
```   342 lemma LIMSEQ_pow:
```
```   343   fixes a :: "'a::{power, real_normed_algebra}"
```
```   344   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
```
```   345 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
```
```   346
```
```   347 lemma LIMSEQ_setsum:
```
```   348   fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   349   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   350   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
```
```   351 using n unfolding LIMSEQ_conv_tendsto by (rule tendsto_setsum)
```
```   352
```
```   353 lemma LIMSEQ_setprod:
```
```   354   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
```
```   355   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   356   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
```
```   357 proof (cases "finite S")
```
```   358   case True
```
```   359   thus ?thesis using n
```
```   360   proof (induct)
```
```   361     case empty
```
```   362     show ?case
```
```   363       by (simp add: LIMSEQ_const)
```
```   364   next
```
```   365     case insert
```
```   366     thus ?case
```
```   367       by (simp add: LIMSEQ_mult)
```
```   368   qed
```
```   369 next
```
```   370   case False
```
```   371   thus ?thesis
```
```   372     by (simp add: setprod_def LIMSEQ_const)
```
```   373 qed
```
```   374
```
```   375 lemma LIMSEQ_add_const:
```
```   376   fixes a :: "'a::real_normed_vector"
```
```   377   shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
```
```   378 by (simp add: LIMSEQ_add LIMSEQ_const)
```
```   379
```
```   380 (* FIXME: delete *)
```
```   381 lemma LIMSEQ_add_minus:
```
```   382   fixes a b :: "'a::real_normed_vector"
```
```   383   shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
```
```   384 by (simp only: LIMSEQ_add LIMSEQ_minus)
```
```   385
```
```   386 lemma LIMSEQ_diff_const:
```
```   387   fixes a b :: "'a::real_normed_vector"
```
```   388   shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
```
```   389 by (simp add: LIMSEQ_diff LIMSEQ_const)
```
```   390
```
```   391 lemma LIMSEQ_diff_approach_zero:
```
```   392   fixes L :: "'a::real_normed_vector"
```
```   393   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```   394 by (drule (1) LIMSEQ_add, simp)
```
```   395
```
```   396 lemma LIMSEQ_diff_approach_zero2:
```
```   397   fixes L :: "'a::real_normed_vector"
```
```   398   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L";
```
```   399 by (drule (1) LIMSEQ_diff, simp)
```
```   400
```
```   401 text{*A sequence tends to zero iff its abs does*}
```
```   402 lemma LIMSEQ_norm_zero:
```
```   403   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   404   shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
```
```   405 by (simp add: LIMSEQ_iff)
```
```   406
```
```   407 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
```
```   408 by (simp add: LIMSEQ_iff)
```
```   409
```
```   410 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
```
```   411 by (drule LIMSEQ_norm, simp)
```
```   412
```
```   413 text{*An unbounded sequence's inverse tends to 0*}
```
```   414
```
```   415 lemma LIMSEQ_inverse_zero:
```
```   416   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   417 apply (rule LIMSEQ_I)
```
```   418 apply (drule_tac x="inverse r" in spec, safe)
```
```   419 apply (rule_tac x="N" in exI, safe)
```
```   420 apply (drule_tac x="n" in spec, safe)
```
```   421 apply (frule positive_imp_inverse_positive)
```
```   422 apply (frule (1) less_imp_inverse_less)
```
```   423 apply (subgoal_tac "0 < X n", simp)
```
```   424 apply (erule (1) order_less_trans)
```
```   425 done
```
```   426
```
```   427 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   428
```
```   429 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   430 apply (rule LIMSEQ_inverse_zero, safe)
```
```   431 apply (cut_tac x = r in reals_Archimedean2)
```
```   432 apply (safe, rule_tac x = n in exI)
```
```   433 apply (auto simp add: real_of_nat_Suc)
```
```   434 done
```
```   435
```
```   436 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   437 infinity is now easily proved*}
```
```   438
```
```   439 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   440      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   441 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   442
```
```   443 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   444      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   445 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   446
```
```   447 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   448      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   449 by (cut_tac b=1 in
```
```   450         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
```
```   451
```
```   452 lemma LIMSEQ_le_const:
```
```   453   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   454 apply (rule ccontr, simp only: linorder_not_le)
```
```   455 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
```
```   456 apply clarsimp
```
```   457 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
```
```   458 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
```
```   459 apply simp
```
```   460 done
```
```   461
```
```   462 lemma LIMSEQ_le_const2:
```
```   463   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   464 apply (subgoal_tac "- a \<le> - x", simp)
```
```   465 apply (rule LIMSEQ_le_const)
```
```   466 apply (erule LIMSEQ_minus)
```
```   467 apply simp
```
```   468 done
```
```   469
```
```   470 lemma LIMSEQ_le:
```
```   471   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
```
```   472 apply (subgoal_tac "0 \<le> y - x", simp)
```
```   473 apply (rule LIMSEQ_le_const)
```
```   474 apply (erule (1) LIMSEQ_diff)
```
```   475 apply (simp add: le_diff_eq)
```
```   476 done
```
```   477
```
```   478
```
```   479 subsection {* Convergence *}
```
```   480
```
```   481 lemma limI: "X ----> L ==> lim X = L"
```
```   482 apply (simp add: lim_def)
```
```   483 apply (blast intro: LIMSEQ_unique)
```
```   484 done
```
```   485
```
```   486 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   487 by (simp add: convergent_def)
```
```   488
```
```   489 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   490 by (auto simp add: convergent_def)
```
```   491
```
```   492 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   493 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   494
```
```   495 lemma convergent_minus_iff:
```
```   496   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   497   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```   498 apply (simp add: convergent_def)
```
```   499 apply (auto dest: LIMSEQ_minus)
```
```   500 apply (drule LIMSEQ_minus, auto)
```
```   501 done
```
```   502
```
```   503 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   504
```
```   505 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   506   unfolding Ex1_def
```
```   507   apply (rule_tac x="nat_rec e f" in exI)
```
```   508   apply (rule conjI)+
```
```   509 apply (rule def_nat_rec_0, simp)
```
```   510 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   511 apply (rule allI, rule impI, rule ext)
```
```   512 apply (erule conjE)
```
```   513 apply (induct_tac x)
```
```   514 apply (simp add: nat_rec_0)
```
```   515 apply (erule_tac x="n" in allE)
```
```   516 apply (simp)
```
```   517 done
```
```   518
```
```   519 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   520
```
```   521 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   522 apply (simp add: subseq_def)
```
```   523 apply (auto dest!: less_imp_Suc_add)
```
```   524 apply (induct_tac k)
```
```   525 apply (auto intro: less_trans)
```
```   526 done
```
```   527
```
```   528 lemma monoseq_Suc:
```
```   529    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
```
```   530                  | (\<forall>n. X (Suc n) \<le> X n))"
```
```   531 apply (simp add: monoseq_def)
```
```   532 apply (auto dest!: le_imp_less_or_eq)
```
```   533 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
```
```   534 apply (induct_tac "ka")
```
```   535 apply (auto intro: order_trans)
```
```   536 apply (erule contrapos_np)
```
```   537 apply (induct_tac "k")
```
```   538 apply (auto intro: order_trans)
```
```   539 done
```
```   540
```
```   541 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```   542 by (simp add: monoseq_def)
```
```   543
```
```   544 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```   545 by (simp add: monoseq_def)
```
```   546
```
```   547 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```   548 by (simp add: monoseq_Suc)
```
```   549
```
```   550 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```   551 by (simp add: monoseq_Suc)
```
```   552
```
```   553 lemma monoseq_minus: assumes "monoseq a"
```
```   554   shows "monoseq (\<lambda> n. - a n)"
```
```   555 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   556   case True
```
```   557   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   558   thus ?thesis by (rule monoI2)
```
```   559 next
```
```   560   case False
```
```   561   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   562   thus ?thesis by (rule monoI1)
```
```   563 qed
```
```   564
```
```   565 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
```
```   566   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
```
```   567          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   568 proof -
```
```   569   { fix x n fix a :: "nat \<Rightarrow> real"
```
```   570     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
```
```   571     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
```
```   572     have "a n \<le> x"
```
```   573     proof (rule ccontr)
```
```   574       assume "\<not> a n \<le> x" hence "x < a n" by auto
```
```   575       hence "0 < a n - x" by auto
```
```   576       from `a ----> x`[THEN LIMSEQ_D, OF this]
```
```   577       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
```
```   578       hence "norm (a (max no n) - x) < a n - x" by auto
```
```   579       moreover
```
```   580       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
```
```   581       hence "x < a (max no n)" by auto
```
```   582       ultimately
```
```   583       have "a (max no n) < a n" by auto
```
```   584       with monotone[where m=n and n="max no n"]
```
```   585       show False by auto
```
```   586     qed
```
```   587   } note top_down = this
```
```   588   { fix x n m fix a :: "nat \<Rightarrow> real"
```
```   589     assume "a ----> x" and "monoseq a" and "a m < x"
```
```   590     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
```
```   591     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   592       case True with top_down and `a ----> x` show ?thesis by auto
```
```   593     next
```
```   594       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
```
```   595       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
```
```   596       hence False using `a m < x` by auto
```
```   597       thus ?thesis ..
```
```   598     qed
```
```   599   } note when_decided = this
```
```   600
```
```   601   show ?thesis
```
```   602   proof (cases "\<exists> m. a m \<noteq> x")
```
```   603     case True then obtain m where "a m \<noteq> x" by auto
```
```   604     show ?thesis
```
```   605     proof (cases "a m < x")
```
```   606       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
```
```   607       show ?thesis by blast
```
```   608     next
```
```   609       case False hence "- a m < - x" using `a m \<noteq> x` by auto
```
```   610       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
```
```   611       show ?thesis by auto
```
```   612     qed
```
```   613   qed auto
```
```   614 qed
```
```   615
```
```   616 text{* for any sequence, there is a mootonic subsequence *}
```
```   617 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   618 proof-
```
```   619   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   620     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   621     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   622     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   623     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   624       using H apply -
```
```   625       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   626       unfolding order_le_less by blast
```
```   627     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   628     {fix n
```
```   629       have "?P (f (Suc n)) (f n)"
```
```   630 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   631 	using H apply -
```
```   632       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   633       unfolding order_le_less by blast
```
```   634     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   635   note fSuc = this
```
```   636     {fix p q assume pq: "p \<ge> f q"
```
```   637       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   638 	by (cases q, simp_all) }
```
```   639     note pqth = this
```
```   640     {fix q
```
```   641       have "f (Suc q) > f q" apply (induct q)
```
```   642 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   643     note fss = this
```
```   644     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   645     {fix a b
```
```   646       have "f a \<le> f (a + b)"
```
```   647       proof(induct b)
```
```   648 	case 0 thus ?case by simp
```
```   649       next
```
```   650 	case (Suc b)
```
```   651 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   652       qed}
```
```   653     note fmon0 = this
```
```   654     have "monoseq (\<lambda>n. s (f n))"
```
```   655     proof-
```
```   656       {fix n
```
```   657 	have "s (f n) \<ge> s (f (Suc n))"
```
```   658 	proof(cases n)
```
```   659 	  case 0
```
```   660 	  assume n0: "n = 0"
```
```   661 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   662 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   663 	next
```
```   664 	  case (Suc m)
```
```   665 	  assume m: "n = Suc m"
```
```   666 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   667 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   668 	qed}
```
```   669       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   670     qed
```
```   671     with th1 have ?thesis by blast}
```
```   672   moreover
```
```   673   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   674     {fix p assume p: "p \<ge> Suc N"
```
```   675       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   676       have "m \<noteq> p" using m(2) by auto
```
```   677       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   678     note th0 = this
```
```   679     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   680     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   681     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   682       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   683     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   684       using N apply -
```
```   685       apply (erule allE[where x="Suc N"], clarsimp)
```
```   686       apply (rule_tac x="m" in exI)
```
```   687       apply auto
```
```   688       apply (subgoal_tac "Suc N \<noteq> m")
```
```   689       apply simp
```
```   690       apply (rule ccontr, simp)
```
```   691       done
```
```   692     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   693     {fix n
```
```   694       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   695 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   696       proof (induct n)
```
```   697 	case 0 thus ?case
```
```   698 	  using f0 N apply auto
```
```   699 	  apply (erule allE[where x="f 0"], clarsimp)
```
```   700 	  apply (rule_tac x="m" in exI, simp)
```
```   701 	  by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   702       next
```
```   703 	case (Suc n)
```
```   704 	from Suc.hyps have Nfn: "N < f n" by blast
```
```   705 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   706 	with Nfn have mN: "m > N" by arith
```
```   707 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   708
```
```   709 	from key have th0: "f (Suc n) > N" by simp
```
```   710 	from N[rule_format, OF th0]
```
```   711 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   712 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   713 	hence "m' > f (Suc n)" using m'(1) by simp
```
```   714 	with key m'(2) show ?case by auto
```
```   715       qed}
```
```   716     note fSuc = this
```
```   717     {fix n
```
```   718       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
```
```   719       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   720     note thf = this
```
```   721     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   722     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   723       apply -
```
```   724       apply (rule disjI1)
```
```   725       apply auto
```
```   726       apply (rule order_less_imp_le)
```
```   727       apply blast
```
```   728       done
```
```   729     then have ?thesis  using sqf by blast}
```
```   730   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   731 qed
```
```   732
```
```   733 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   734 proof(induct n)
```
```   735   case 0 thus ?case by simp
```
```   736 next
```
```   737   case (Suc n)
```
```   738   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   739   have "n < f (Suc n)" by arith
```
```   740   thus ?case by arith
```
```   741 qed
```
```   742
```
```   743 lemma LIMSEQ_subseq_LIMSEQ:
```
```   744   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```   745 apply (auto simp add: LIMSEQ_def)
```
```   746 apply (drule_tac x=r in spec, clarify)
```
```   747 apply (rule_tac x=no in exI, clarify)
```
```   748 apply (blast intro: seq_suble le_trans dest!: spec)
```
```   749 done
```
```   750
```
```   751 subsection {* Bounded Monotonic Sequences *}
```
```   752
```
```   753
```
```   754 text{*Bounded Sequence*}
```
```   755
```
```   756 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   757 by (simp add: Bseq_def)
```
```   758
```
```   759 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   760 by (auto simp add: Bseq_def)
```
```   761
```
```   762 lemma lemma_NBseq_def:
```
```   763      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
```
```   764       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   765 apply auto
```
```   766  prefer 2 apply force
```
```   767 apply (cut_tac x = K in reals_Archimedean2, clarify)
```
```   768 apply (rule_tac x = n in exI, clarify)
```
```   769 apply (drule_tac x = na in spec)
```
```   770 apply (auto simp add: real_of_nat_Suc)
```
```   771 done
```
```   772
```
```   773 text{* alternative definition for Bseq *}
```
```   774 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   775 apply (simp add: Bseq_def)
```
```   776 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   777 done
```
```   778
```
```   779 lemma lemma_NBseq_def2:
```
```   780      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   781 apply (subst lemma_NBseq_def, auto)
```
```   782 apply (rule_tac x = "Suc N" in exI)
```
```   783 apply (rule_tac [2] x = N in exI)
```
```   784 apply (auto simp add: real_of_nat_Suc)
```
```   785  prefer 2 apply (blast intro: order_less_imp_le)
```
```   786 apply (drule_tac x = n in spec, simp)
```
```   787 done
```
```   788
```
```   789 (* yet another definition for Bseq *)
```
```   790 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   791 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   792
```
```   793 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   794
```
```   795 lemma Bseq_isUb:
```
```   796   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   797 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   798
```
```   799
```
```   800 text{* Use completeness of reals (supremum property)
```
```   801    to show that any bounded sequence has a least upper bound*}
```
```   802
```
```   803 lemma Bseq_isLub:
```
```   804   "!!(X::nat=>real). Bseq X ==>
```
```   805    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   806 by (blast intro: reals_complete Bseq_isUb)
```
```   807
```
```   808 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   809
```
```   810 lemma lemma_converg1:
```
```   811      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
```
```   812                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
```
```   813                |] ==> \<forall>n \<ge> ma. X n = X ma"
```
```   814 apply safe
```
```   815 apply (drule_tac y = "X n" in isLubD2)
```
```   816 apply (blast dest: order_antisym)+
```
```   817 done
```
```   818
```
```   819 text{* The best of both worlds: Easier to prove this result as a standard
```
```   820    theorem and then use equivalence to "transfer" it into the
```
```   821    equivalent nonstandard form if needed!*}
```
```   822
```
```   823 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   824 apply (simp add: LIMSEQ_def)
```
```   825 apply (rule_tac x = "X m" in exI, safe)
```
```   826 apply (rule_tac x = m in exI, safe)
```
```   827 apply (drule spec, erule impE, auto)
```
```   828 done
```
```   829
```
```   830 lemma lemma_converg2:
```
```   831    "!!(X::nat=>real).
```
```   832     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
```
```   833 apply safe
```
```   834 apply (drule_tac y = "X m" in isLubD2)
```
```   835 apply (auto dest!: order_le_imp_less_or_eq)
```
```   836 done
```
```   837
```
```   838 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
```
```   839 by (rule setleI [THEN isUbI], auto)
```
```   840
```
```   841 text{* FIXME: @{term "U - T < U"} is redundant *}
```
```   842 lemma lemma_converg4: "!!(X::nat=> real).
```
```   843                [| \<forall>m. X m ~= U;
```
```   844                   isLub UNIV {x. \<exists>n. X n = x} U;
```
```   845                   0 < T;
```
```   846                   U + - T < U
```
```   847                |] ==> \<exists>m. U + -T < X m & X m < U"
```
```   848 apply (drule lemma_converg2, assumption)
```
```   849 apply (rule ccontr, simp)
```
```   850 apply (simp add: linorder_not_less)
```
```   851 apply (drule lemma_converg3)
```
```   852 apply (drule isLub_le_isUb, assumption)
```
```   853 apply (auto dest: order_less_le_trans)
```
```   854 done
```
```   855
```
```   856 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```   857
```
```   858 lemma Bseq_mono_convergent:
```
```   859      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
```
```   860 apply (simp add: convergent_def)
```
```   861 apply (frule Bseq_isLub, safe)
```
```   862 apply (case_tac "\<exists>m. X m = U", auto)
```
```   863 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
```
```   864 (* second case *)
```
```   865 apply (rule_tac x = U in exI)
```
```   866 apply (subst LIMSEQ_iff, safe)
```
```   867 apply (frule lemma_converg2, assumption)
```
```   868 apply (drule lemma_converg4, auto)
```
```   869 apply (rule_tac x = m in exI, safe)
```
```   870 apply (subgoal_tac "X m \<le> X n")
```
```   871  prefer 2 apply blast
```
```   872 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
```
```   873 done
```
```   874
```
```   875 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```   876 by (simp add: Bseq_def)
```
```   877
```
```   878 text{*Main monotonicity theorem*}
```
```   879 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
```
```   880 apply (simp add: monoseq_def, safe)
```
```   881 apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
```
```   882 apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
```
```   883 apply (auto intro!: Bseq_mono_convergent)
```
```   884 done
```
```   885
```
```   886 subsubsection{*Increasing and Decreasing Series*}
```
```   887
```
```   888 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```   889   by (simp add: incseq_def monoseq_def)
```
```   890
```
```   891 lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
```
```   892   using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
```
```   893 proof
```
```   894   assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
```
```   895   thus ?thesis by simp
```
```   896 next
```
```   897   assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
```
```   898   hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
```
```   899     by (auto simp add: incseq_def intro: order_antisym)
```
```   900   have X: "!!n. X n = X 0"
```
```   901     by (blast intro: const [of 0])
```
```   902   have "X = (\<lambda>n. X 0)"
```
```   903     by (blast intro: ext X)
```
```   904   hence "L = X 0" using LIMSEQ_const [of "X 0"]
```
```   905     by (auto intro: LIMSEQ_unique lim)
```
```   906   thus ?thesis
```
```   907     by (blast intro: eq_refl X)
```
```   908 qed
```
```   909
```
```   910 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```   911   by (simp add: decseq_def monoseq_def)
```
```   912
```
```   913 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
```
```   914   by (simp add: decseq_def incseq_def)
```
```   915
```
```   916
```
```   917 lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
```
```   918 proof -
```
```   919   have inc: "incseq (\<lambda>n. - X n)" using dec
```
```   920     by (simp add: decseq_eq_incseq)
```
```   921   have "- X n \<le> - L"
```
```   922     by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim)
```
```   923   thus ?thesis
```
```   924     by simp
```
```   925 qed
```
```   926
```
```   927 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```   928
```
```   929 text{*alternative formulation for boundedness*}
```
```   930 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```   931 apply (unfold Bseq_def, safe)
```
```   932 apply (rule_tac [2] x = "k + norm x" in exI)
```
```   933 apply (rule_tac x = K in exI, simp)
```
```   934 apply (rule exI [where x = 0], auto)
```
```   935 apply (erule order_less_le_trans, simp)
```
```   936 apply (drule_tac x=n in spec, fold diff_def)
```
```   937 apply (drule order_trans [OF norm_triangle_ineq2])
```
```   938 apply simp
```
```   939 done
```
```   940
```
```   941 text{*alternative formulation for boundedness*}
```
```   942 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```   943 apply safe
```
```   944 apply (simp add: Bseq_def, safe)
```
```   945 apply (rule_tac x = "K + norm (X N)" in exI)
```
```   946 apply auto
```
```   947 apply (erule order_less_le_trans, simp)
```
```   948 apply (rule_tac x = N in exI, safe)
```
```   949 apply (drule_tac x = n in spec)
```
```   950 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```   951 apply (auto simp add: Bseq_iff2)
```
```   952 done
```
```   953
```
```   954 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```   955 apply (simp add: Bseq_def)
```
```   956 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```   957 apply (drule_tac x = n in spec, arith)
```
```   958 done
```
```   959
```
```   960
```
```   961 subsection {* Cauchy Sequences *}
```
```   962
```
```   963 lemma metric_CauchyI:
```
```   964   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```   965 by (simp add: Cauchy_def)
```
```   966
```
```   967 lemma metric_CauchyD:
```
```   968   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```   969 by (simp add: Cauchy_def)
```
```   970
```
```   971 lemma Cauchy_iff:
```
```   972   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   973   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```   974 unfolding Cauchy_def dist_norm ..
```
```   975
```
```   976 lemma CauchyI:
```
```   977   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   978   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"
```
```   979 by (simp add: Cauchy_iff)
```
```   980
```
```   981 lemma CauchyD:
```
```   982   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   983   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```   984 by (simp add: Cauchy_iff)
```
```   985
```
```   986 lemma Cauchy_subseq_Cauchy:
```
```   987   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```   988 apply (auto simp add: Cauchy_def)
```
```   989 apply (drule_tac x=e in spec, clarify)
```
```   990 apply (rule_tac x=M in exI, clarify)
```
```   991 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```   992 done
```
```   993
```
```   994 subsubsection {* Cauchy Sequences are Bounded *}
```
```   995
```
```   996 text{*A Cauchy sequence is bounded -- this is the standard
```
```   997   proof mechanization rather than the nonstandard proof*}
```
```   998
```
```   999 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1000           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1001 apply (clarify, drule spec, drule (1) mp)
```
```  1002 apply (simp only: norm_minus_commute)
```
```  1003 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1004 apply simp
```
```  1005 done
```
```  1006
```
```  1007 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```  1008 apply (simp add: Cauchy_iff)
```
```  1009 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```  1010 apply (drule_tac x="M" in spec, simp)
```
```  1011 apply (drule lemmaCauchy)
```
```  1012 apply (rule_tac k="M" in Bseq_offset)
```
```  1013 apply (simp add: Bseq_def)
```
```  1014 apply (rule_tac x="1 + norm (X M)" in exI)
```
```  1015 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```  1016 apply (simp add: order_less_imp_le)
```
```  1017 done
```
```  1018
```
```  1019 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1020
```
```  1021 axclass complete_space \<subseteq> metric_space
```
```  1022   Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1023
```
```  1024 axclass banach \<subseteq> real_normed_vector, complete_space
```
```  1025
```
```  1026 theorem LIMSEQ_imp_Cauchy:
```
```  1027   assumes X: "X ----> a" shows "Cauchy X"
```
```  1028 proof (rule metric_CauchyI)
```
```  1029   fix e::real assume "0 < e"
```
```  1030   hence "0 < e/2" by simp
```
```  1031   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```  1032   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```  1033   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```  1034   proof (intro exI allI impI)
```
```  1035     fix m assume "N \<le> m"
```
```  1036     hence m: "dist (X m) a < e/2" using N by fast
```
```  1037     fix n assume "N \<le> n"
```
```  1038     hence n: "dist (X n) a < e/2" using N by fast
```
```  1039     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```  1040       by (rule dist_triangle2)
```
```  1041     also from m n have "\<dots> < e" by simp
```
```  1042     finally show "dist (X m) (X n) < e" .
```
```  1043   qed
```
```  1044 qed
```
```  1045
```
```  1046 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1047 unfolding convergent_def
```
```  1048 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1049
```
```  1050 lemma Cauchy_convergent_iff:
```
```  1051   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1052   shows "Cauchy X = convergent X"
```
```  1053 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1054
```
```  1055 lemma convergent_subseq_convergent:
```
```  1056   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1057   shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```  1058   by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
```
```  1059
```
```  1060 text {*
```
```  1061 Proof that Cauchy sequences converge based on the one from
```
```  1062 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1063 *}
```
```  1064
```
```  1065 text {*
```
```  1066   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1067   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1068 *}
```
```  1069
```
```  1070 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1071 by (simp add: isUbI setleI)
```
```  1072
```
```  1073 lemma real_abs_diff_less_iff:
```
```  1074   "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
```
```  1075 by auto
```
```  1076
```
```  1077 locale real_Cauchy =
```
```  1078   fixes X :: "nat \<Rightarrow> real"
```
```  1079   assumes X: "Cauchy X"
```
```  1080   fixes S :: "real set"
```
```  1081   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1082
```
```  1083 lemma real_CauchyI:
```
```  1084   assumes "Cauchy X"
```
```  1085   shows "real_Cauchy X"
```
```  1086   proof qed (fact assms)
```
```  1087
```
```  1088 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
```
```  1089 by (unfold S_def, auto)
```
```  1090
```
```  1091 lemma (in real_Cauchy) bound_isUb:
```
```  1092   assumes N: "\<forall>n\<ge>N. X n < x"
```
```  1093   shows "isUb UNIV S x"
```
```  1094 proof (rule isUb_UNIV_I)
```
```  1095   fix y::real assume "y \<in> S"
```
```  1096   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1097     by (simp add: S_def)
```
```  1098   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1099   hence "y < X (max M N)" by simp
```
```  1100   also have "\<dots> < x" using N by simp
```
```  1101   finally show "y \<le> x"
```
```  1102     by (rule order_less_imp_le)
```
```  1103 qed
```
```  1104
```
```  1105 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
```
```  1106 proof (rule reals_complete)
```
```  1107   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```  1108     using CauchyD [OF X zero_less_one] by fast
```
```  1109   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```  1110   show "\<exists>x. x \<in> S"
```
```  1111   proof
```
```  1112     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1113       by (simp add: real_abs_diff_less_iff)
```
```  1114     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1115   qed
```
```  1116   show "\<exists>u. isUb UNIV S u"
```
```  1117   proof
```
```  1118     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1119       by (simp add: real_abs_diff_less_iff)
```
```  1120     thus "isUb UNIV S (X N + 1)"
```
```  1121       by (rule bound_isUb)
```
```  1122   qed
```
```  1123 qed
```
```  1124
```
```  1125 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
```
```  1126   assumes x: "isLub UNIV S x"
```
```  1127   shows "X ----> x"
```
```  1128 proof (rule LIMSEQ_I)
```
```  1129   fix r::real assume "0 < r"
```
```  1130   hence r: "0 < r/2" by simp
```
```  1131   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```  1132     using CauchyD [OF X r] by fast
```
```  1133   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```  1134   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1135     by (simp only: real_norm_def real_abs_diff_less_iff)
```
```  1136
```
```  1137   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1138   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1139   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1140
```
```  1141   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1142   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1143   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1144
```
```  1145   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```  1146   proof (intro exI allI impI)
```
```  1147     fix n assume n: "N \<le> n"
```
```  1148     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1149     thus "norm (X n - x) < r" using 1 2
```
```  1150       by (simp add: real_abs_diff_less_iff)
```
```  1151   qed
```
```  1152 qed
```
```  1153
```
```  1154 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
```
```  1155 proof -
```
```  1156   obtain x where "isLub UNIV S x"
```
```  1157     using isLub_ex by fast
```
```  1158   hence "X ----> x"
```
```  1159     by (rule isLub_imp_LIMSEQ)
```
```  1160   thus ?thesis ..
```
```  1161 qed
```
```  1162
```
```  1163 lemma real_Cauchy_convergent:
```
```  1164   fixes X :: "nat \<Rightarrow> real"
```
```  1165   shows "Cauchy X \<Longrightarrow> convergent X"
```
```  1166 unfolding convergent_def
```
```  1167 by (rule real_Cauchy.LIMSEQ_ex)
```
```  1168  (rule real_CauchyI)
```
```  1169
```
```  1170 instance real :: banach
```
```  1171 by intro_classes (rule real_Cauchy_convergent)
```
```  1172
```
```  1173
```
```  1174 subsection {* Power Sequences *}
```
```  1175
```
```  1176 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1177 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1178   also fact that bounded and monotonic sequence converges.*}
```
```  1179
```
```  1180 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1181 apply (simp add: Bseq_def)
```
```  1182 apply (rule_tac x = 1 in exI)
```
```  1183 apply (simp add: power_abs)
```
```  1184 apply (auto dest: power_mono)
```
```  1185 done
```
```  1186
```
```  1187 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1188 apply (clarify intro!: mono_SucI2)
```
```  1189 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1190 done
```
```  1191
```
```  1192 lemma convergent_realpow:
```
```  1193   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1194 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1195
```
```  1196 lemma LIMSEQ_inverse_realpow_zero_lemma:
```
```  1197   fixes x :: real
```
```  1198   assumes x: "0 \<le> x"
```
```  1199   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1200 apply (induct n)
```
```  1201 apply simp
```
```  1202 apply simp
```
```  1203 apply (rule order_trans)
```
```  1204 prefer 2
```
```  1205 apply (erule mult_left_mono)
```
```  1206 apply (rule add_increasing [OF x], simp)
```
```  1207 apply (simp add: real_of_nat_Suc)
```
```  1208 apply (simp add: ring_distribs)
```
```  1209 apply (simp add: mult_nonneg_nonneg x)
```
```  1210 done
```
```  1211
```
```  1212 lemma LIMSEQ_inverse_realpow_zero:
```
```  1213   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1214 proof (rule LIMSEQ_inverse_zero [rule_format])
```
```  1215   fix y :: real
```
```  1216   assume x: "1 < x"
```
```  1217   hence "0 < x - 1" by simp
```
```  1218   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
```
```  1219     by (rule reals_Archimedean3)
```
```  1220   hence "\<exists>N::nat. y < real N * (x - 1)" ..
```
```  1221   then obtain N::nat where "y < real N * (x - 1)" ..
```
```  1222   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
```
```  1223   also have "\<dots> \<le> (x - 1 + 1) ^ N"
```
```  1224     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
```
```  1225   also have "\<dots> = x ^ N" by simp
```
```  1226   finally have "y < x ^ N" .
```
```  1227   hence "\<forall>n\<ge>N. y < x ^ n"
```
```  1228     apply clarify
```
```  1229     apply (erule order_less_le_trans)
```
```  1230     apply (erule power_increasing)
```
```  1231     apply (rule order_less_imp_le [OF x])
```
```  1232     done
```
```  1233   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
```
```  1234 qed
```
```  1235
```
```  1236 lemma LIMSEQ_realpow_zero:
```
```  1237   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1238 proof (cases)
```
```  1239   assume "x = 0"
```
```  1240   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
```
```  1241   thus ?thesis by (rule LIMSEQ_imp_Suc)
```
```  1242 next
```
```  1243   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1244   hence x0: "0 < x" by simp
```
```  1245   assume x1: "x < 1"
```
```  1246   from x0 x1 have "1 < inverse x"
```
```  1247     by (rule real_inverse_gt_one)
```
```  1248   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1249     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1250   thus ?thesis by (simp add: power_inverse)
```
```  1251 qed
```
```  1252
```
```  1253 lemma LIMSEQ_power_zero:
```
```  1254   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1255   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1256 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1257 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
```
```  1258 apply (simp add: power_abs norm_power_ineq)
```
```  1259 done
```
```  1260
```
```  1261 lemma LIMSEQ_divide_realpow_zero:
```
```  1262   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
```
```  1263 apply (cut_tac a = a and x1 = "inverse x" in
```
```  1264         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
```
```  1265 apply (auto simp add: divide_inverse power_inverse)
```
```  1266 apply (simp add: inverse_eq_divide pos_divide_less_eq)
```
```  1267 done
```
```  1268
```
```  1269 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1270
```
```  1271 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
```
```  1272 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1273
```
```  1274 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
```
```  1275 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
```
```  1276 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
```
```  1277 done
```
```  1278
```
```  1279 end
```