src/HOL/SEQ.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30273 ecd6f0ca62ea child 30730 4d3565f2cb0e permissions -rw-r--r--
simplified method setup;
```     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 RealVector RComplete
```
```    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 => 'a::real_normed_vector, 'a] => bool"
```
```    22     ("((_)/ ----> (_))" [60, 60] 60) where
```
```    23     --{*Standard definition of convergence of sequence*}
```
```    24   [code del]: "X ----> L = (\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X n - L) < r))"
```
```    25
```
```    26 definition
```
```    27   lim :: "(nat => 'a::real_normed_vector) => 'a" where
```
```    28     --{*Standard definition of limit using choice operator*}
```
```    29   "lim X = (THE L. X ----> L)"
```
```    30
```
```    31 definition
```
```    32   convergent :: "(nat => 'a::real_normed_vector) => 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 for monotonicity*}
```
```    44   [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))"
```
```    45
```
```    46 definition
```
```    47   subseq :: "(nat => nat) => bool" where
```
```    48     --{*Definition of subsequence*}
```
```    49   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
```
```    50
```
```    51 definition
```
```    52   Cauchy :: "(nat => 'a::real_normed_vector) => bool" where
```
```    53     --{*Standard definition of the Cauchy condition*}
```
```    54   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. norm (X m - X n) < e)"
```
```    55
```
```    56
```
```    57 subsection {* Bounded Sequences *}
```
```    58
```
```    59 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
```
```    60 unfolding Bseq_def
```
```    61 proof (intro exI conjI allI)
```
```    62   show "0 < max K 1" by simp
```
```    63 next
```
```    64   fix n::nat
```
```    65   have "norm (X n) \<le> K" by (rule K)
```
```    66   thus "norm (X n) \<le> max K 1" by simp
```
```    67 qed
```
```    68
```
```    69 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```    70 unfolding Bseq_def by auto
```
```    71
```
```    72 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
```
```    73 proof (rule BseqI')
```
```    74   let ?A = "norm ` X ` {..N}"
```
```    75   have 1: "finite ?A" by simp
```
```    76   fix n::nat
```
```    77   show "norm (X n) \<le> max K (Max ?A)"
```
```    78   proof (cases rule: linorder_le_cases)
```
```    79     assume "n \<ge> N"
```
```    80     hence "norm (X n) \<le> K" using K by simp
```
```    81     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```    82   next
```
```    83     assume "n \<le> N"
```
```    84     hence "norm (X n) \<in> ?A" by simp
```
```    85     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
```
```    86     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```    87   qed
```
```    88 qed
```
```    89
```
```    90 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```    91 unfolding Bseq_def by auto
```
```    92
```
```    93 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```    94 apply (erule BseqE)
```
```    95 apply (rule_tac N="k" and K="K" in BseqI2')
```
```    96 apply clarify
```
```    97 apply (drule_tac x="n - k" in spec, simp)
```
```    98 done
```
```    99
```
```   100
```
```   101 subsection {* Sequences That Converge to Zero *}
```
```   102
```
```   103 lemma ZseqI:
```
```   104   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
```
```   105 unfolding Zseq_def by simp
```
```   106
```
```   107 lemma ZseqD:
```
```   108   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
```
```   109 unfolding Zseq_def by simp
```
```   110
```
```   111 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
```
```   112 unfolding Zseq_def by simp
```
```   113
```
```   114 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
```
```   115 unfolding Zseq_def by force
```
```   116
```
```   117 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
```
```   118 unfolding Zseq_def by simp
```
```   119
```
```   120 lemma Zseq_imp_Zseq:
```
```   121   assumes X: "Zseq X"
```
```   122   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
```
```   123   shows "Zseq (\<lambda>n. Y n)"
```
```   124 proof (cases)
```
```   125   assume K: "0 < K"
```
```   126   show ?thesis
```
```   127   proof (rule ZseqI)
```
```   128     fix r::real assume "0 < r"
```
```   129     hence "0 < r / K"
```
```   130       using K by (rule divide_pos_pos)
```
```   131     then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
```
```   132       using ZseqD [OF X] by fast
```
```   133     hence "\<forall>n\<ge>N. norm (X n) * K < r"
```
```   134       by (simp add: pos_less_divide_eq K)
```
```   135     hence "\<forall>n\<ge>N. norm (Y n) < r"
```
```   136       by (simp add: order_le_less_trans [OF Y])
```
```   137     thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
```
```   138   qed
```
```   139 next
```
```   140   assume "\<not> 0 < K"
```
```   141   hence K: "K \<le> 0" by (simp only: linorder_not_less)
```
```   142   {
```
```   143     fix n::nat
```
```   144     have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
```
```   145     also have "\<dots> \<le> norm (X n) * 0"
```
```   146       using K norm_ge_zero by (rule mult_left_mono)
```
```   147     finally have "norm (Y n) = 0" by simp
```
```   148   }
```
```   149   thus ?thesis by (simp add: Zseq_zero)
```
```   150 qed
```
```   151
```
```   152 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
```
```   153 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
```
```   154
```
```   155 lemma Zseq_add:
```
```   156   assumes X: "Zseq X"
```
```   157   assumes Y: "Zseq Y"
```
```   158   shows "Zseq (\<lambda>n. X n + Y n)"
```
```   159 proof (rule ZseqI)
```
```   160   fix r::real assume "0 < r"
```
```   161   hence r: "0 < r / 2" by simp
```
```   162   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
```
```   163     using ZseqD [OF X r] by fast
```
```   164   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
```
```   165     using ZseqD [OF Y r] by fast
```
```   166   show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
```
```   167   proof (intro exI allI impI)
```
```   168     fix n assume n: "max M N \<le> n"
```
```   169     have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
```
```   170       by (rule norm_triangle_ineq)
```
```   171     also have "\<dots> < r/2 + r/2"
```
```   172     proof (rule add_strict_mono)
```
```   173       from M n show "norm (X n) < r/2" by simp
```
```   174       from N n show "norm (Y n) < r/2" by simp
```
```   175     qed
```
```   176     finally show "norm (X n + Y n) < r" by simp
```
```   177   qed
```
```   178 qed
```
```   179
```
```   180 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
```
```   181 unfolding Zseq_def by simp
```
```   182
```
```   183 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
```
```   184 by (simp only: diff_minus Zseq_add Zseq_minus)
```
```   185
```
```   186 lemma (in bounded_linear) Zseq:
```
```   187   assumes X: "Zseq X"
```
```   188   shows "Zseq (\<lambda>n. f (X n))"
```
```   189 proof -
```
```   190   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   191     using bounded by fast
```
```   192   with X show ?thesis
```
```   193     by (rule Zseq_imp_Zseq)
```
```   194 qed
```
```   195
```
```   196 lemma (in bounded_bilinear) Zseq:
```
```   197   assumes X: "Zseq X"
```
```   198   assumes Y: "Zseq Y"
```
```   199   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   200 proof (rule ZseqI)
```
```   201   fix r::real assume r: "0 < r"
```
```   202   obtain K where K: "0 < K"
```
```   203     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   204     using pos_bounded by fast
```
```   205   from K have K': "0 < inverse K"
```
```   206     by (rule positive_imp_inverse_positive)
```
```   207   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
```
```   208     using ZseqD [OF X r] by fast
```
```   209   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
```
```   210     using ZseqD [OF Y K'] by fast
```
```   211   show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
```
```   212   proof (intro exI allI impI)
```
```   213     fix n assume n: "max M N \<le> n"
```
```   214     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   215       by (rule norm_le)
```
```   216     also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
```
```   217     proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
```
```   218       from M n show Xn: "norm (X n) < r" by simp
```
```   219       from N n show Yn: "norm (Y n) < inverse K" by simp
```
```   220     qed
```
```   221     also from K have "r * inverse K * K = r" by simp
```
```   222     finally show "norm (X n ** Y n) < r" .
```
```   223   qed
```
```   224 qed
```
```   225
```
```   226 lemma (in bounded_bilinear) Zseq_prod_Bseq:
```
```   227   assumes X: "Zseq X"
```
```   228   assumes Y: "Bseq Y"
```
```   229   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   230 proof -
```
```   231   obtain K where K: "0 \<le> K"
```
```   232     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   233     using nonneg_bounded by fast
```
```   234   obtain B where B: "0 < B"
```
```   235     and norm_Y: "\<And>n. norm (Y n) \<le> B"
```
```   236     using Y [unfolded Bseq_def] by fast
```
```   237   from X show ?thesis
```
```   238   proof (rule Zseq_imp_Zseq)
```
```   239     fix n::nat
```
```   240     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   241       by (rule norm_le)
```
```   242     also have "\<dots> \<le> norm (X n) * B * K"
```
```   243       by (intro mult_mono' order_refl norm_Y norm_ge_zero
```
```   244                 mult_nonneg_nonneg K)
```
```   245     also have "\<dots> = norm (X n) * (B * K)"
```
```   246       by (rule mult_assoc)
```
```   247     finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
```
```   248   qed
```
```   249 qed
```
```   250
```
```   251 lemma (in bounded_bilinear) Bseq_prod_Zseq:
```
```   252   assumes X: "Bseq X"
```
```   253   assumes Y: "Zseq Y"
```
```   254   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   255 proof -
```
```   256   obtain K where K: "0 \<le> K"
```
```   257     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   258     using nonneg_bounded by fast
```
```   259   obtain B where B: "0 < B"
```
```   260     and norm_X: "\<And>n. norm (X n) \<le> B"
```
```   261     using X [unfolded Bseq_def] by fast
```
```   262   from Y show ?thesis
```
```   263   proof (rule Zseq_imp_Zseq)
```
```   264     fix n::nat
```
```   265     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   266       by (rule norm_le)
```
```   267     also have "\<dots> \<le> B * norm (Y n) * K"
```
```   268       by (intro mult_mono' order_refl norm_X norm_ge_zero
```
```   269                 mult_nonneg_nonneg K)
```
```   270     also have "\<dots> = norm (Y n) * (B * K)"
```
```   271       by (simp only: mult_ac)
```
```   272     finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
```
```   273   qed
```
```   274 qed
```
```   275
```
```   276 lemma (in bounded_bilinear) Zseq_left:
```
```   277   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
```
```   278 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
```
```   279
```
```   280 lemma (in bounded_bilinear) Zseq_right:
```
```   281   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
```
```   282 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
```
```   283
```
```   284 lemmas Zseq_mult = mult.Zseq
```
```   285 lemmas Zseq_mult_right = mult.Zseq_right
```
```   286 lemmas Zseq_mult_left = mult.Zseq_left
```
```   287
```
```   288
```
```   289 subsection {* Limits of Sequences *}
```
```   290
```
```   291 lemma LIMSEQ_iff:
```
```   292       "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   293 by (rule LIMSEQ_def)
```
```   294
```
```   295 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
```
```   296 by (simp only: LIMSEQ_def Zseq_def)
```
```   297
```
```   298 lemma LIMSEQ_I:
```
```   299   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   300 by (simp add: LIMSEQ_def)
```
```   301
```
```   302 lemma LIMSEQ_D:
```
```   303   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   304 by (simp add: LIMSEQ_def)
```
```   305
```
```   306 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
```
```   307 by (simp add: LIMSEQ_def)
```
```   308
```
```   309 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l = (k = l)"
```
```   310 by (simp add: LIMSEQ_Zseq_iff Zseq_const_iff)
```
```   311
```
```   312 lemma LIMSEQ_norm: "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
```
```   313 apply (simp add: LIMSEQ_def, safe)
```
```   314 apply (drule_tac x="r" in spec, safe)
```
```   315 apply (rule_tac x="no" in exI, safe)
```
```   316 apply (drule_tac x="n" in spec, safe)
```
```   317 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
```
```   318 done
```
```   319
```
```   320 lemma LIMSEQ_ignore_initial_segment:
```
```   321   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   322 apply (rule LIMSEQ_I)
```
```   323 apply (drule (1) LIMSEQ_D)
```
```   324 apply (erule exE, rename_tac N)
```
```   325 apply (rule_tac x=N in exI)
```
```   326 apply simp
```
```   327 done
```
```   328
```
```   329 lemma LIMSEQ_offset:
```
```   330   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   331 apply (rule LIMSEQ_I)
```
```   332 apply (drule (1) LIMSEQ_D)
```
```   333 apply (erule exE, rename_tac N)
```
```   334 apply (rule_tac x="N + k" in exI)
```
```   335 apply clarify
```
```   336 apply (drule_tac x="n - k" in spec)
```
```   337 apply (simp add: le_diff_conv2)
```
```   338 done
```
```   339
```
```   340 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   341 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   342
```
```   343 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   344 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   345
```
```   346 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   347 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   348
```
```   349 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   350   unfolding LIMSEQ_def
```
```   351   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   352
```
```   353
```
```   354 lemma add_diff_add:
```
```   355   fixes a b c d :: "'a::ab_group_add"
```
```   356   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```   357 by simp
```
```   358
```
```   359 lemma minus_diff_minus:
```
```   360   fixes a b :: "'a::ab_group_add"
```
```   361   shows "(- a) - (- b) = - (a - b)"
```
```   362 by simp
```
```   363
```
```   364 lemma LIMSEQ_add: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
```
```   365 by (simp only: LIMSEQ_Zseq_iff add_diff_add Zseq_add)
```
```   366
```
```   367 lemma LIMSEQ_minus: "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
```
```   368 by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
```
```   369
```
```   370 lemma LIMSEQ_minus_cancel: "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
```
```   371 by (drule LIMSEQ_minus, simp)
```
```   372
```
```   373 lemma LIMSEQ_diff: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
```
```   374 by (simp add: diff_minus LIMSEQ_add LIMSEQ_minus)
```
```   375
```
```   376 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   377 by (drule (1) LIMSEQ_diff, simp add: LIMSEQ_const_iff)
```
```   378
```
```   379 lemma (in bounded_linear) LIMSEQ:
```
```   380   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```
```   381 by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
```
```   382
```
```   383 lemma (in bounded_bilinear) LIMSEQ:
```
```   384   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
```
```   385 by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
```
```   386                Zseq_add Zseq Zseq_left Zseq_right)
```
```   387
```
```   388 lemma LIMSEQ_mult:
```
```   389   fixes a b :: "'a::real_normed_algebra"
```
```   390   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
```
```   391 by (rule mult.LIMSEQ)
```
```   392
```
```   393 lemma inverse_diff_inverse:
```
```   394   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   395    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   396 by (simp add: algebra_simps)
```
```   397
```
```   398 lemma Bseq_inverse_lemma:
```
```   399   fixes x :: "'a::real_normed_div_algebra"
```
```   400   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   401 apply (subst nonzero_norm_inverse, clarsimp)
```
```   402 apply (erule (1) le_imp_inverse_le)
```
```   403 done
```
```   404
```
```   405 lemma Bseq_inverse:
```
```   406   fixes a :: "'a::real_normed_div_algebra"
```
```   407   assumes X: "X ----> a"
```
```   408   assumes a: "a \<noteq> 0"
```
```   409   shows "Bseq (\<lambda>n. inverse (X n))"
```
```   410 proof -
```
```   411   from a have "0 < norm a" by simp
```
```   412   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   413   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   414   obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
```
```   415     using LIMSEQ_D [OF X r1] by fast
```
```   416   show ?thesis
```
```   417   proof (rule BseqI2' [rule_format])
```
```   418     fix n assume n: "N \<le> n"
```
```   419     hence 1: "norm (X n - a) < r" by (rule N)
```
```   420     hence 2: "X n \<noteq> 0" using r2 by auto
```
```   421     hence "norm (inverse (X n)) = inverse (norm (X n))"
```
```   422       by (rule nonzero_norm_inverse)
```
```   423     also have "\<dots> \<le> inverse (norm a - r)"
```
```   424     proof (rule le_imp_inverse_le)
```
```   425       show "0 < norm a - r" using r2 by simp
```
```   426     next
```
```   427       have "norm a - norm (X n) \<le> norm (a - X n)"
```
```   428         by (rule norm_triangle_ineq2)
```
```   429       also have "\<dots> = norm (X n - a)"
```
```   430         by (rule norm_minus_commute)
```
```   431       also have "\<dots> < r" using 1 .
```
```   432       finally show "norm a - r \<le> norm (X n)" by simp
```
```   433     qed
```
```   434     finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
```
```   435   qed
```
```   436 qed
```
```   437
```
```   438 lemma LIMSEQ_inverse_lemma:
```
```   439   fixes a :: "'a::real_normed_div_algebra"
```
```   440   shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
```
```   441          \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
```
```   442 apply (subst LIMSEQ_Zseq_iff)
```
```   443 apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
```
```   444 apply (rule Zseq_minus)
```
```   445 apply (rule Zseq_mult_left)
```
```   446 apply (rule mult.Bseq_prod_Zseq)
```
```   447 apply (erule (1) Bseq_inverse)
```
```   448 apply (simp add: LIMSEQ_Zseq_iff)
```
```   449 done
```
```   450
```
```   451 lemma LIMSEQ_inverse:
```
```   452   fixes a :: "'a::real_normed_div_algebra"
```
```   453   assumes X: "X ----> a"
```
```   454   assumes a: "a \<noteq> 0"
```
```   455   shows "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   456 proof -
```
```   457   from a have "0 < norm a" by simp
```
```   458   then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
```
```   459     using LIMSEQ_D [OF X] by fast
```
```   460   hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
```
```   461   hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
```
```   462
```
```   463   from X have "(\<lambda>n. X (n + k)) ----> a"
```
```   464     by (rule LIMSEQ_ignore_initial_segment)
```
```   465   hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
```
```   466     using a k by (rule LIMSEQ_inverse_lemma)
```
```   467   thus "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   468     by (rule LIMSEQ_offset)
```
```   469 qed
```
```   470
```
```   471 lemma LIMSEQ_divide:
```
```   472   fixes a b :: "'a::real_normed_field"
```
```   473   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
```
```   474 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
```
```   475
```
```   476 lemma LIMSEQ_pow:
```
```   477   fixes a :: "'a::{real_normed_algebra,recpower}"
```
```   478   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
```
```   479 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
```
```   480
```
```   481 lemma LIMSEQ_setsum:
```
```   482   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   483   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
```
```   484 proof (cases "finite S")
```
```   485   case True
```
```   486   thus ?thesis using n
```
```   487   proof (induct)
```
```   488     case empty
```
```   489     show ?case
```
```   490       by (simp add: LIMSEQ_const)
```
```   491   next
```
```   492     case insert
```
```   493     thus ?case
```
```   494       by (simp add: LIMSEQ_add)
```
```   495   qed
```
```   496 next
```
```   497   case False
```
```   498   thus ?thesis
```
```   499     by (simp add: LIMSEQ_const)
```
```   500 qed
```
```   501
```
```   502 lemma LIMSEQ_setprod:
```
```   503   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
```
```   504   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   505   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
```
```   506 proof (cases "finite S")
```
```   507   case True
```
```   508   thus ?thesis using n
```
```   509   proof (induct)
```
```   510     case empty
```
```   511     show ?case
```
```   512       by (simp add: LIMSEQ_const)
```
```   513   next
```
```   514     case insert
```
```   515     thus ?case
```
```   516       by (simp add: LIMSEQ_mult)
```
```   517   qed
```
```   518 next
```
```   519   case False
```
```   520   thus ?thesis
```
```   521     by (simp add: setprod_def LIMSEQ_const)
```
```   522 qed
```
```   523
```
```   524 lemma LIMSEQ_add_const: "f ----> a ==> (%n.(f n + b)) ----> a + b"
```
```   525 by (simp add: LIMSEQ_add LIMSEQ_const)
```
```   526
```
```   527 (* FIXME: delete *)
```
```   528 lemma LIMSEQ_add_minus:
```
```   529      "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
```
```   530 by (simp only: LIMSEQ_add LIMSEQ_minus)
```
```   531
```
```   532 lemma LIMSEQ_diff_const: "f ----> a ==> (%n.(f n  - b)) ----> a - b"
```
```   533 by (simp add: LIMSEQ_diff LIMSEQ_const)
```
```   534
```
```   535 lemma LIMSEQ_diff_approach_zero:
```
```   536   "g ----> L ==> (%x. f x - g x) ----> 0  ==>
```
```   537      f ----> L"
```
```   538   apply (drule LIMSEQ_add)
```
```   539   apply assumption
```
```   540   apply simp
```
```   541 done
```
```   542
```
```   543 lemma LIMSEQ_diff_approach_zero2:
```
```   544   "f ----> L ==> (%x. f x - g x) ----> 0  ==>
```
```   545      g ----> L";
```
```   546   apply (drule LIMSEQ_diff)
```
```   547   apply assumption
```
```   548   apply simp
```
```   549 done
```
```   550
```
```   551 text{*A sequence tends to zero iff its abs does*}
```
```   552 lemma LIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----> 0) = (X ----> 0)"
```
```   553 by (simp add: LIMSEQ_def)
```
```   554
```
```   555 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
```
```   556 by (simp add: LIMSEQ_def)
```
```   557
```
```   558 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
```
```   559 by (drule LIMSEQ_norm, simp)
```
```   560
```
```   561 text{*An unbounded sequence's inverse tends to 0*}
```
```   562
```
```   563 lemma LIMSEQ_inverse_zero:
```
```   564   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   565 apply (rule LIMSEQ_I)
```
```   566 apply (drule_tac x="inverse r" in spec, safe)
```
```   567 apply (rule_tac x="N" in exI, safe)
```
```   568 apply (drule_tac x="n" in spec, safe)
```
```   569 apply (frule positive_imp_inverse_positive)
```
```   570 apply (frule (1) less_imp_inverse_less)
```
```   571 apply (subgoal_tac "0 < X n", simp)
```
```   572 apply (erule (1) order_less_trans)
```
```   573 done
```
```   574
```
```   575 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   576
```
```   577 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   578 apply (rule LIMSEQ_inverse_zero, safe)
```
```   579 apply (cut_tac x = r in reals_Archimedean2)
```
```   580 apply (safe, rule_tac x = n in exI)
```
```   581 apply (auto simp add: real_of_nat_Suc)
```
```   582 done
```
```   583
```
```   584 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   585 infinity is now easily proved*}
```
```   586
```
```   587 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   588      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   589 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   590
```
```   591 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   592      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   593 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   594
```
```   595 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   596      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   597 by (cut_tac b=1 in
```
```   598         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
```
```   599
```
```   600 lemma LIMSEQ_le_const:
```
```   601   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   602 apply (rule ccontr, simp only: linorder_not_le)
```
```   603 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
```
```   604 apply clarsimp
```
```   605 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
```
```   606 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
```
```   607 apply simp
```
```   608 done
```
```   609
```
```   610 lemma LIMSEQ_le_const2:
```
```   611   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   612 apply (subgoal_tac "- a \<le> - x", simp)
```
```   613 apply (rule LIMSEQ_le_const)
```
```   614 apply (erule LIMSEQ_minus)
```
```   615 apply simp
```
```   616 done
```
```   617
```
```   618 lemma LIMSEQ_le:
```
```   619   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
```
```   620 apply (subgoal_tac "0 \<le> y - x", simp)
```
```   621 apply (rule LIMSEQ_le_const)
```
```   622 apply (erule (1) LIMSEQ_diff)
```
```   623 apply (simp add: le_diff_eq)
```
```   624 done
```
```   625
```
```   626
```
```   627 subsection {* Convergence *}
```
```   628
```
```   629 lemma limI: "X ----> L ==> lim X = L"
```
```   630 apply (simp add: lim_def)
```
```   631 apply (blast intro: LIMSEQ_unique)
```
```   632 done
```
```   633
```
```   634 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   635 by (simp add: convergent_def)
```
```   636
```
```   637 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   638 by (auto simp add: convergent_def)
```
```   639
```
```   640 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   641 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   642
```
```   643 lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
```
```   644 apply (simp add: convergent_def)
```
```   645 apply (auto dest: LIMSEQ_minus)
```
```   646 apply (drule LIMSEQ_minus, auto)
```
```   647 done
```
```   648
```
```   649 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   650
```
```   651 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   652   unfolding Ex1_def
```
```   653   apply (rule_tac x="nat_rec e f" in exI)
```
```   654   apply (rule conjI)+
```
```   655 apply (rule def_nat_rec_0, simp)
```
```   656 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   657 apply (rule allI, rule impI, rule ext)
```
```   658 apply (erule conjE)
```
```   659 apply (induct_tac x)
```
```   660 apply (simp add: nat_rec_0)
```
```   661 apply (erule_tac x="n" in allE)
```
```   662 apply (simp)
```
```   663 done
```
```   664
```
```   665 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   666
```
```   667 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   668 apply (simp add: subseq_def)
```
```   669 apply (auto dest!: less_imp_Suc_add)
```
```   670 apply (induct_tac k)
```
```   671 apply (auto intro: less_trans)
```
```   672 done
```
```   673
```
```   674 lemma monoseq_Suc:
```
```   675    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
```
```   676                  | (\<forall>n. X (Suc n) \<le> X n))"
```
```   677 apply (simp add: monoseq_def)
```
```   678 apply (auto dest!: le_imp_less_or_eq)
```
```   679 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
```
```   680 apply (induct_tac "ka")
```
```   681 apply (auto intro: order_trans)
```
```   682 apply (erule contrapos_np)
```
```   683 apply (induct_tac "k")
```
```   684 apply (auto intro: order_trans)
```
```   685 done
```
```   686
```
```   687 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```   688 by (simp add: monoseq_def)
```
```   689
```
```   690 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```   691 by (simp add: monoseq_def)
```
```   692
```
```   693 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```   694 by (simp add: monoseq_Suc)
```
```   695
```
```   696 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```   697 by (simp add: monoseq_Suc)
```
```   698
```
```   699 lemma monoseq_minus: assumes "monoseq a"
```
```   700   shows "monoseq (\<lambda> n. - a n)"
```
```   701 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   702   case True
```
```   703   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   704   thus ?thesis by (rule monoI2)
```
```   705 next
```
```   706   case False
```
```   707   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   708   thus ?thesis by (rule monoI1)
```
```   709 qed
```
```   710
```
```   711 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
```
```   712   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
```
```   713          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   714 proof -
```
```   715   { fix x n fix a :: "nat \<Rightarrow> real"
```
```   716     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
```
```   717     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
```
```   718     have "a n \<le> x"
```
```   719     proof (rule ccontr)
```
```   720       assume "\<not> a n \<le> x" hence "x < a n" by auto
```
```   721       hence "0 < a n - x" by auto
```
```   722       from `a ----> x`[THEN LIMSEQ_D, OF this]
```
```   723       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
```
```   724       hence "norm (a (max no n) - x) < a n - x" by auto
```
```   725       moreover
```
```   726       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
```
```   727       hence "x < a (max no n)" by auto
```
```   728       ultimately
```
```   729       have "a (max no n) < a n" by auto
```
```   730       with monotone[where m=n and n="max no n"]
```
```   731       show False by auto
```
```   732     qed
```
```   733   } note top_down = this
```
```   734   { fix x n m fix a :: "nat \<Rightarrow> real"
```
```   735     assume "a ----> x" and "monoseq a" and "a m < x"
```
```   736     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
```
```   737     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   738       case True with top_down and `a ----> x` show ?thesis by auto
```
```   739     next
```
```   740       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
```
```   741       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
```
```   742       hence False using `a m < x` by auto
```
```   743       thus ?thesis ..
```
```   744     qed
```
```   745   } note when_decided = this
```
```   746
```
```   747   show ?thesis
```
```   748   proof (cases "\<exists> m. a m \<noteq> x")
```
```   749     case True then obtain m where "a m \<noteq> x" by auto
```
```   750     show ?thesis
```
```   751     proof (cases "a m < x")
```
```   752       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
```
```   753       show ?thesis by blast
```
```   754     next
```
```   755       case False hence "- a m < - x" using `a m \<noteq> x` by auto
```
```   756       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
```
```   757       show ?thesis by auto
```
```   758     qed
```
```   759   qed auto
```
```   760 qed
```
```   761
```
```   762 text{* for any sequence, there is a mootonic subsequence *}
```
```   763 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   764 proof-
```
```   765   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   766     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   767     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   768     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   769     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   770       using H apply -
```
```   771       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   772       unfolding order_le_less by blast
```
```   773     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   774     {fix n
```
```   775       have "?P (f (Suc n)) (f n)"
```
```   776 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   777 	using H apply -
```
```   778       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   779       unfolding order_le_less by blast
```
```   780     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   781   note fSuc = this
```
```   782     {fix p q assume pq: "p \<ge> f q"
```
```   783       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   784 	by (cases q, simp_all) }
```
```   785     note pqth = this
```
```   786     {fix q
```
```   787       have "f (Suc q) > f q" apply (induct q)
```
```   788 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   789     note fss = this
```
```   790     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   791     {fix a b
```
```   792       have "f a \<le> f (a + b)"
```
```   793       proof(induct b)
```
```   794 	case 0 thus ?case by simp
```
```   795       next
```
```   796 	case (Suc b)
```
```   797 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   798       qed}
```
```   799     note fmon0 = this
```
```   800     have "monoseq (\<lambda>n. s (f n))"
```
```   801     proof-
```
```   802       {fix n
```
```   803 	have "s (f n) \<ge> s (f (Suc n))"
```
```   804 	proof(cases n)
```
```   805 	  case 0
```
```   806 	  assume n0: "n = 0"
```
```   807 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   808 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   809 	next
```
```   810 	  case (Suc m)
```
```   811 	  assume m: "n = Suc m"
```
```   812 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   813 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   814 	qed}
```
```   815       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   816     qed
```
```   817     with th1 have ?thesis by blast}
```
```   818   moreover
```
```   819   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   820     {fix p assume p: "p \<ge> Suc N"
```
```   821       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   822       have "m \<noteq> p" using m(2) by auto
```
```   823       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   824     note th0 = this
```
```   825     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   826     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   827     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   828       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   829     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   830       using N apply -
```
```   831       apply (erule allE[where x="Suc N"], clarsimp)
```
```   832       apply (rule_tac x="m" in exI)
```
```   833       apply auto
```
```   834       apply (subgoal_tac "Suc N \<noteq> m")
```
```   835       apply simp
```
```   836       apply (rule ccontr, simp)
```
```   837       done
```
```   838     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   839     {fix n
```
```   840       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   841 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   842       proof (induct n)
```
```   843 	case 0 thus ?case
```
```   844 	  using f0 N apply auto
```
```   845 	  apply (erule allE[where x="f 0"], clarsimp)
```
```   846 	  apply (rule_tac x="m" in exI, simp)
```
```   847 	  by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   848       next
```
```   849 	case (Suc n)
```
```   850 	from Suc.hyps have Nfn: "N < f n" by blast
```
```   851 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   852 	with Nfn have mN: "m > N" by arith
```
```   853 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   854
```
```   855 	from key have th0: "f (Suc n) > N" by simp
```
```   856 	from N[rule_format, OF th0]
```
```   857 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   858 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   859 	hence "m' > f (Suc n)" using m'(1) by simp
```
```   860 	with key m'(2) show ?case by auto
```
```   861       qed}
```
```   862     note fSuc = this
```
```   863     {fix n
```
```   864       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
```
```   865       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   866     note thf = this
```
```   867     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   868     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   869       apply -
```
```   870       apply (rule disjI1)
```
```   871       apply auto
```
```   872       apply (rule order_less_imp_le)
```
```   873       apply blast
```
```   874       done
```
```   875     then have ?thesis  using sqf by blast}
```
```   876   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   877 qed
```
```   878
```
```   879 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   880 proof(induct n)
```
```   881   case 0 thus ?case by simp
```
```   882 next
```
```   883   case (Suc n)
```
```   884   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   885   have "n < f (Suc n)" by arith
```
```   886   thus ?case by arith
```
```   887 qed
```
```   888
```
```   889 subsection {* Bounded Monotonic Sequences *}
```
```   890
```
```   891
```
```   892 text{*Bounded Sequence*}
```
```   893
```
```   894 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   895 by (simp add: Bseq_def)
```
```   896
```
```   897 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   898 by (auto simp add: Bseq_def)
```
```   899
```
```   900 lemma lemma_NBseq_def:
```
```   901      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
```
```   902       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   903 apply auto
```
```   904  prefer 2 apply force
```
```   905 apply (cut_tac x = K in reals_Archimedean2, clarify)
```
```   906 apply (rule_tac x = n in exI, clarify)
```
```   907 apply (drule_tac x = na in spec)
```
```   908 apply (auto simp add: real_of_nat_Suc)
```
```   909 done
```
```   910
```
```   911 text{* alternative definition for Bseq *}
```
```   912 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   913 apply (simp add: Bseq_def)
```
```   914 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   915 done
```
```   916
```
```   917 lemma lemma_NBseq_def2:
```
```   918      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   919 apply (subst lemma_NBseq_def, auto)
```
```   920 apply (rule_tac x = "Suc N" in exI)
```
```   921 apply (rule_tac  x = N in exI)
```
```   922 apply (auto simp add: real_of_nat_Suc)
```
```   923  prefer 2 apply (blast intro: order_less_imp_le)
```
```   924 apply (drule_tac x = n in spec, simp)
```
```   925 done
```
```   926
```
```   927 (* yet another definition for Bseq *)
```
```   928 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   929 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   930
```
```   931 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   932
```
```   933 lemma Bseq_isUb:
```
```   934   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   935 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   936
```
```   937
```
```   938 text{* Use completeness of reals (supremum property)
```
```   939    to show that any bounded sequence has a least upper bound*}
```
```   940
```
```   941 lemma Bseq_isLub:
```
```   942   "!!(X::nat=>real). Bseq X ==>
```
```   943    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   944 by (blast intro: reals_complete Bseq_isUb)
```
```   945
```
```   946 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   947
```
```   948 lemma lemma_converg1:
```
```   949      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
```
```   950                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
```
```   951                |] ==> \<forall>n \<ge> ma. X n = X ma"
```
```   952 apply safe
```
```   953 apply (drule_tac y = "X n" in isLubD2)
```
```   954 apply (blast dest: order_antisym)+
```
```   955 done
```
```   956
```
```   957 text{* The best of both worlds: Easier to prove this result as a standard
```
```   958    theorem and then use equivalence to "transfer" it into the
```
```   959    equivalent nonstandard form if needed!*}
```
```   960
```
```   961 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   962 apply (simp add: LIMSEQ_def)
```
```   963 apply (rule_tac x = "X m" in exI, safe)
```
```   964 apply (rule_tac x = m in exI, safe)
```
```   965 apply (drule spec, erule impE, auto)
```
```   966 done
```
```   967
```
```   968 lemma lemma_converg2:
```
```   969    "!!(X::nat=>real).
```
```   970     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
```
```   971 apply safe
```
```   972 apply (drule_tac y = "X m" in isLubD2)
```
```   973 apply (auto dest!: order_le_imp_less_or_eq)
```
```   974 done
```
```   975
```
```   976 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
```
```   977 by (rule setleI [THEN isUbI], auto)
```
```   978
```
```   979 text{* FIXME: @{term "U - T < U"} is redundant *}
```
```   980 lemma lemma_converg4: "!!(X::nat=> real).
```
```   981                [| \<forall>m. X m ~= U;
```
```   982                   isLub UNIV {x. \<exists>n. X n = x} U;
```
```   983                   0 < T;
```
```   984                   U + - T < U
```
```   985                |] ==> \<exists>m. U + -T < X m & X m < U"
```
```   986 apply (drule lemma_converg2, assumption)
```
```   987 apply (rule ccontr, simp)
```
```   988 apply (simp add: linorder_not_less)
```
```   989 apply (drule lemma_converg3)
```
```   990 apply (drule isLub_le_isUb, assumption)
```
```   991 apply (auto dest: order_less_le_trans)
```
```   992 done
```
```   993
```
```   994 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```   995
```
```   996 lemma Bseq_mono_convergent:
```
```   997      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
```
```   998 apply (simp add: convergent_def)
```
```   999 apply (frule Bseq_isLub, safe)
```
```  1000 apply (case_tac "\<exists>m. X m = U", auto)
```
```  1001 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
```
```  1002 (* second case *)
```
```  1003 apply (rule_tac x = U in exI)
```
```  1004 apply (subst LIMSEQ_iff, safe)
```
```  1005 apply (frule lemma_converg2, assumption)
```
```  1006 apply (drule lemma_converg4, auto)
```
```  1007 apply (rule_tac x = m in exI, safe)
```
```  1008 apply (subgoal_tac "X m \<le> X n")
```
```  1009  prefer 2 apply blast
```
```  1010 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
```
```  1011 done
```
```  1012
```
```  1013 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```  1014 by (simp add: Bseq_def)
```
```  1015
```
```  1016 text{*Main monotonicity theorem*}
```
```  1017 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
```
```  1018 apply (simp add: monoseq_def, safe)
```
```  1019 apply (rule_tac  convergent_minus_iff [THEN ssubst])
```
```  1020 apply (drule_tac  Bseq_minus_iff [THEN ssubst])
```
```  1021 apply (auto intro!: Bseq_mono_convergent)
```
```  1022 done
```
```  1023
```
```  1024 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```  1025
```
```  1026 text{*alternative formulation for boundedness*}
```
```  1027 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```  1028 apply (unfold Bseq_def, safe)
```
```  1029 apply (rule_tac  x = "k + norm x" in exI)
```
```  1030 apply (rule_tac x = K in exI, simp)
```
```  1031 apply (rule exI [where x = 0], auto)
```
```  1032 apply (erule order_less_le_trans, simp)
```
```  1033 apply (drule_tac x=n in spec, fold diff_def)
```
```  1034 apply (drule order_trans [OF norm_triangle_ineq2])
```
```  1035 apply simp
```
```  1036 done
```
```  1037
```
```  1038 text{*alternative formulation for boundedness*}
```
```  1039 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```  1040 apply safe
```
```  1041 apply (simp add: Bseq_def, safe)
```
```  1042 apply (rule_tac x = "K + norm (X N)" in exI)
```
```  1043 apply auto
```
```  1044 apply (erule order_less_le_trans, simp)
```
```  1045 apply (rule_tac x = N in exI, safe)
```
```  1046 apply (drule_tac x = n in spec)
```
```  1047 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```  1048 apply (auto simp add: Bseq_iff2)
```
```  1049 done
```
```  1050
```
```  1051 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```  1052 apply (simp add: Bseq_def)
```
```  1053 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```  1054 apply (drule_tac x = n in spec, arith)
```
```  1055 done
```
```  1056
```
```  1057
```
```  1058 subsection {* Cauchy Sequences *}
```
```  1059
```
```  1060 lemma CauchyI:
```
```  1061   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
```
```  1062 by (simp add: Cauchy_def)
```
```  1063
```
```  1064 lemma CauchyD:
```
```  1065   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1066 by (simp add: Cauchy_def)
```
```  1067
```
```  1068 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1069
```
```  1070 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1071   proof mechanization rather than the nonstandard proof*}
```
```  1072
```
```  1073 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1074           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1075 apply (clarify, drule spec, drule (1) mp)
```
```  1076 apply (simp only: norm_minus_commute)
```
```  1077 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1078 apply simp
```
```  1079 done
```
```  1080
```
```  1081 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```  1082 apply (simp add: Cauchy_def)
```
```  1083 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```  1084 apply (drule_tac x="M" in spec, simp)
```
```  1085 apply (drule lemmaCauchy)
```
```  1086 apply (rule_tac k="M" in Bseq_offset)
```
```  1087 apply (simp add: Bseq_def)
```
```  1088 apply (rule_tac x="1 + norm (X M)" in exI)
```
```  1089 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```  1090 apply (simp add: order_less_imp_le)
```
```  1091 done
```
```  1092
```
```  1093 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1094
```
```  1095 axclass banach \<subseteq> real_normed_vector
```
```  1096   Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1097
```
```  1098 theorem LIMSEQ_imp_Cauchy:
```
```  1099   assumes X: "X ----> a" shows "Cauchy X"
```
```  1100 proof (rule CauchyI)
```
```  1101   fix e::real assume "0 < e"
```
```  1102   hence "0 < e/2" by simp
```
```  1103   with X have "\<exists>N. \<forall>n\<ge>N. norm (X n - a) < e/2" by (rule LIMSEQ_D)
```
```  1104   then obtain N where N: "\<forall>n\<ge>N. norm (X n - a) < e/2" ..
```
```  1105   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < e"
```
```  1106   proof (intro exI allI impI)
```
```  1107     fix m assume "N \<le> m"
```
```  1108     hence m: "norm (X m - a) < e/2" using N by fast
```
```  1109     fix n assume "N \<le> n"
```
```  1110     hence n: "norm (X n - a) < e/2" using N by fast
```
```  1111     have "norm (X m - X n) = norm ((X m - a) - (X n - a))" by simp
```
```  1112     also have "\<dots> \<le> norm (X m - a) + norm (X n - a)"
```
```  1113       by (rule norm_triangle_ineq4)
```
```  1114     also from m n have "\<dots> < e" by(simp add:field_simps)
```
```  1115     finally show "norm (X m - X n) < e" .
```
```  1116   qed
```
```  1117 qed
```
```  1118
```
```  1119 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1120 unfolding convergent_def
```
```  1121 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1122
```
```  1123 text {*
```
```  1124 Proof that Cauchy sequences converge based on the one from
```
```  1125 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1126 *}
```
```  1127
```
```  1128 text {*
```
```  1129   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1130   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1131 *}
```
```  1132
```
```  1133 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1134 by (simp add: isUbI setleI)
```
```  1135
```
```  1136 lemma real_abs_diff_less_iff:
```
```  1137   "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
```
```  1138 by auto
```
```  1139
```
```  1140 locale real_Cauchy =
```
```  1141   fixes X :: "nat \<Rightarrow> real"
```
```  1142   assumes X: "Cauchy X"
```
```  1143   fixes S :: "real set"
```
```  1144   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1145
```
```  1146 lemma real_CauchyI:
```
```  1147   assumes "Cauchy X"
```
```  1148   shows "real_Cauchy X"
```
```  1149   proof qed (fact assms)
```
```  1150
```
```  1151 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
```
```  1152 by (unfold S_def, auto)
```
```  1153
```
```  1154 lemma (in real_Cauchy) bound_isUb:
```
```  1155   assumes N: "\<forall>n\<ge>N. X n < x"
```
```  1156   shows "isUb UNIV S x"
```
```  1157 proof (rule isUb_UNIV_I)
```
```  1158   fix y::real assume "y \<in> S"
```
```  1159   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1160     by (simp add: S_def)
```
```  1161   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1162   hence "y < X (max M N)" by simp
```
```  1163   also have "\<dots> < x" using N by simp
```
```  1164   finally show "y \<le> x"
```
```  1165     by (rule order_less_imp_le)
```
```  1166 qed
```
```  1167
```
```  1168 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
```
```  1169 proof (rule reals_complete)
```
```  1170   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```  1171     using CauchyD [OF X zero_less_one] by fast
```
```  1172   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```  1173   show "\<exists>x. x \<in> S"
```
```  1174   proof
```
```  1175     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1176       by (simp add: real_abs_diff_less_iff)
```
```  1177     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1178   qed
```
```  1179   show "\<exists>u. isUb UNIV S u"
```
```  1180   proof
```
```  1181     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1182       by (simp add: real_abs_diff_less_iff)
```
```  1183     thus "isUb UNIV S (X N + 1)"
```
```  1184       by (rule bound_isUb)
```
```  1185   qed
```
```  1186 qed
```
```  1187
```
```  1188 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
```
```  1189   assumes x: "isLub UNIV S x"
```
```  1190   shows "X ----> x"
```
```  1191 proof (rule LIMSEQ_I)
```
```  1192   fix r::real assume "0 < r"
```
```  1193   hence r: "0 < r/2" by simp
```
```  1194   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```  1195     using CauchyD [OF X r] by fast
```
```  1196   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```  1197   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1198     by (simp only: real_norm_def real_abs_diff_less_iff)
```
```  1199
```
```  1200   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1201   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1202   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1203
```
```  1204   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1205   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1206   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1207
```
```  1208   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```  1209   proof (intro exI allI impI)
```
```  1210     fix n assume n: "N \<le> n"
```
```  1211     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1212     thus "norm (X n - x) < r" using 1 2
```
```  1213       by (simp add: real_abs_diff_less_iff)
```
```  1214   qed
```
```  1215 qed
```
```  1216
```
```  1217 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
```
```  1218 proof -
```
```  1219   obtain x where "isLub UNIV S x"
```
```  1220     using isLub_ex by fast
```
```  1221   hence "X ----> x"
```
```  1222     by (rule isLub_imp_LIMSEQ)
```
```  1223   thus ?thesis ..
```
```  1224 qed
```
```  1225
```
```  1226 lemma real_Cauchy_convergent:
```
```  1227   fixes X :: "nat \<Rightarrow> real"
```
```  1228   shows "Cauchy X \<Longrightarrow> convergent X"
```
```  1229 unfolding convergent_def
```
```  1230 by (rule real_Cauchy.LIMSEQ_ex)
```
```  1231  (rule real_CauchyI)
```
```  1232
```
```  1233 instance real :: banach
```
```  1234 by intro_classes (rule real_Cauchy_convergent)
```
```  1235
```
```  1236 lemma Cauchy_convergent_iff:
```
```  1237   fixes X :: "nat \<Rightarrow> 'a::banach"
```
```  1238   shows "Cauchy X = convergent X"
```
```  1239 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1240
```
```  1241
```
```  1242 subsection {* Power Sequences *}
```
```  1243
```
```  1244 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1245 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1246   also fact that bounded and monotonic sequence converges.*}
```
```  1247
```
```  1248 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1249 apply (simp add: Bseq_def)
```
```  1250 apply (rule_tac x = 1 in exI)
```
```  1251 apply (simp add: power_abs)
```
```  1252 apply (auto dest: power_mono)
```
```  1253 done
```
```  1254
```
```  1255 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1256 apply (clarify intro!: mono_SucI2)
```
```  1257 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1258 done
```
```  1259
```
```  1260 lemma convergent_realpow:
```
```  1261   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1262 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1263
```
```  1264 lemma LIMSEQ_inverse_realpow_zero_lemma:
```
```  1265   fixes x :: real
```
```  1266   assumes x: "0 \<le> x"
```
```  1267   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1268 apply (induct n)
```
```  1269 apply simp
```
```  1270 apply simp
```
```  1271 apply (rule order_trans)
```
```  1272 prefer 2
```
```  1273 apply (erule mult_left_mono)
```
```  1274 apply (rule add_increasing [OF x], simp)
```
```  1275 apply (simp add: real_of_nat_Suc)
```
```  1276 apply (simp add: ring_distribs)
```
```  1277 apply (simp add: mult_nonneg_nonneg x)
```
```  1278 done
```
```  1279
```
```  1280 lemma LIMSEQ_inverse_realpow_zero:
```
```  1281   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1282 proof (rule LIMSEQ_inverse_zero [rule_format])
```
```  1283   fix y :: real
```
```  1284   assume x: "1 < x"
```
```  1285   hence "0 < x - 1" by simp
```
```  1286   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
```
```  1287     by (rule reals_Archimedean3)
```
```  1288   hence "\<exists>N::nat. y < real N * (x - 1)" ..
```
```  1289   then obtain N::nat where "y < real N * (x - 1)" ..
```
```  1290   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
```
```  1291   also have "\<dots> \<le> (x - 1 + 1) ^ N"
```
```  1292     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
```
```  1293   also have "\<dots> = x ^ N" by simp
```
```  1294   finally have "y < x ^ N" .
```
```  1295   hence "\<forall>n\<ge>N. y < x ^ n"
```
```  1296     apply clarify
```
```  1297     apply (erule order_less_le_trans)
```
```  1298     apply (erule power_increasing)
```
```  1299     apply (rule order_less_imp_le [OF x])
```
```  1300     done
```
```  1301   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
```
```  1302 qed
```
```  1303
```
```  1304 lemma LIMSEQ_realpow_zero:
```
```  1305   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1306 proof (cases)
```
```  1307   assume "x = 0"
```
```  1308   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
```
```  1309   thus ?thesis by (rule LIMSEQ_imp_Suc)
```
```  1310 next
```
```  1311   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1312   hence x0: "0 < x" by simp
```
```  1313   assume x1: "x < 1"
```
```  1314   from x0 x1 have "1 < inverse x"
```
```  1315     by (rule real_inverse_gt_one)
```
```  1316   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1317     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1318   thus ?thesis by (simp add: power_inverse)
```
```  1319 qed
```
```  1320
```
```  1321 lemma LIMSEQ_power_zero:
```
```  1322   fixes x :: "'a::{real_normed_algebra_1,recpower}"
```
```  1323   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1324 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1325 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
```
```  1326 apply (simp add: power_abs norm_power_ineq)
```
```  1327 done
```
```  1328
```
```  1329 lemma LIMSEQ_divide_realpow_zero:
```
```  1330   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
```
```  1331 apply (cut_tac a = a and x1 = "inverse x" in
```
```  1332         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
```
```  1333 apply (auto simp add: divide_inverse power_inverse)
```
```  1334 apply (simp add: inverse_eq_divide pos_divide_less_eq)
```
```  1335 done
```
```  1336
```
```  1337 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1338
```
```  1339 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
```
```  1340 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1341
```
```  1342 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
```
```  1343 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
```
```  1344 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
```
```  1345 done
```
```  1346
```
```  1347 end
```