src/HOL/SEQ.thy
 author haftmann Mon Apr 27 10:11:44 2009 +0200 (2009-04-27) changeset 31001 7e6ffd8f51a9 parent 30730 4d3565f2cb0e child 31017 2c227493ea56 permissions -rw-r--r--
cleaned up theory power further
```     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 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 => 'a::real_normed_vector) => 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. norm (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
```
```   114 subsection {* Sequences That Converge to Zero *}
```
```   115
```
```   116 lemma ZseqI:
```
```   117   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
```
```   118 unfolding Zseq_def by simp
```
```   119
```
```   120 lemma ZseqD:
```
```   121   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
```
```   122 unfolding Zseq_def by simp
```
```   123
```
```   124 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
```
```   125 unfolding Zseq_def by simp
```
```   126
```
```   127 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
```
```   128 unfolding Zseq_def by force
```
```   129
```
```   130 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
```
```   131 unfolding Zseq_def by simp
```
```   132
```
```   133 lemma Zseq_imp_Zseq:
```
```   134   assumes X: "Zseq X"
```
```   135   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
```
```   136   shows "Zseq (\<lambda>n. Y n)"
```
```   137 proof (cases)
```
```   138   assume K: "0 < K"
```
```   139   show ?thesis
```
```   140   proof (rule ZseqI)
```
```   141     fix r::real assume "0 < r"
```
```   142     hence "0 < r / K"
```
```   143       using K by (rule divide_pos_pos)
```
```   144     then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
```
```   145       using ZseqD [OF X] by fast
```
```   146     hence "\<forall>n\<ge>N. norm (X n) * K < r"
```
```   147       by (simp add: pos_less_divide_eq K)
```
```   148     hence "\<forall>n\<ge>N. norm (Y n) < r"
```
```   149       by (simp add: order_le_less_trans [OF Y])
```
```   150     thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
```
```   151   qed
```
```   152 next
```
```   153   assume "\<not> 0 < K"
```
```   154   hence K: "K \<le> 0" by (simp only: linorder_not_less)
```
```   155   {
```
```   156     fix n::nat
```
```   157     have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
```
```   158     also have "\<dots> \<le> norm (X n) * 0"
```
```   159       using K norm_ge_zero by (rule mult_left_mono)
```
```   160     finally have "norm (Y n) = 0" by simp
```
```   161   }
```
```   162   thus ?thesis by (simp add: Zseq_zero)
```
```   163 qed
```
```   164
```
```   165 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
```
```   166 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
```
```   167
```
```   168 lemma Zseq_add:
```
```   169   assumes X: "Zseq X"
```
```   170   assumes Y: "Zseq Y"
```
```   171   shows "Zseq (\<lambda>n. X n + Y n)"
```
```   172 proof (rule ZseqI)
```
```   173   fix r::real assume "0 < r"
```
```   174   hence r: "0 < r / 2" by simp
```
```   175   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
```
```   176     using ZseqD [OF X r] by fast
```
```   177   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
```
```   178     using ZseqD [OF Y r] by fast
```
```   179   show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
```
```   180   proof (intro exI allI impI)
```
```   181     fix n assume n: "max M N \<le> n"
```
```   182     have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
```
```   183       by (rule norm_triangle_ineq)
```
```   184     also have "\<dots> < r/2 + r/2"
```
```   185     proof (rule add_strict_mono)
```
```   186       from M n show "norm (X n) < r/2" by simp
```
```   187       from N n show "norm (Y n) < r/2" by simp
```
```   188     qed
```
```   189     finally show "norm (X n + Y n) < r" by simp
```
```   190   qed
```
```   191 qed
```
```   192
```
```   193 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
```
```   194 unfolding Zseq_def by simp
```
```   195
```
```   196 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
```
```   197 by (simp only: diff_minus Zseq_add Zseq_minus)
```
```   198
```
```   199 lemma (in bounded_linear) Zseq:
```
```   200   assumes X: "Zseq X"
```
```   201   shows "Zseq (\<lambda>n. f (X n))"
```
```   202 proof -
```
```   203   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   204     using bounded by fast
```
```   205   with X show ?thesis
```
```   206     by (rule Zseq_imp_Zseq)
```
```   207 qed
```
```   208
```
```   209 lemma (in bounded_bilinear) Zseq:
```
```   210   assumes X: "Zseq X"
```
```   211   assumes Y: "Zseq Y"
```
```   212   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   213 proof (rule ZseqI)
```
```   214   fix r::real assume r: "0 < r"
```
```   215   obtain K where K: "0 < K"
```
```   216     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   217     using pos_bounded by fast
```
```   218   from K have K': "0 < inverse K"
```
```   219     by (rule positive_imp_inverse_positive)
```
```   220   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
```
```   221     using ZseqD [OF X r] by fast
```
```   222   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
```
```   223     using ZseqD [OF Y K'] by fast
```
```   224   show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
```
```   225   proof (intro exI allI impI)
```
```   226     fix n assume n: "max M N \<le> n"
```
```   227     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   228       by (rule norm_le)
```
```   229     also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
```
```   230     proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
```
```   231       from M n show Xn: "norm (X n) < r" by simp
```
```   232       from N n show Yn: "norm (Y n) < inverse K" by simp
```
```   233     qed
```
```   234     also from K have "r * inverse K * K = r" by simp
```
```   235     finally show "norm (X n ** Y n) < r" .
```
```   236   qed
```
```   237 qed
```
```   238
```
```   239 lemma (in bounded_bilinear) Zseq_prod_Bseq:
```
```   240   assumes X: "Zseq X"
```
```   241   assumes Y: "Bseq Y"
```
```   242   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   243 proof -
```
```   244   obtain K where K: "0 \<le> K"
```
```   245     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   246     using nonneg_bounded by fast
```
```   247   obtain B where B: "0 < B"
```
```   248     and norm_Y: "\<And>n. norm (Y n) \<le> B"
```
```   249     using Y [unfolded Bseq_def] by fast
```
```   250   from X show ?thesis
```
```   251   proof (rule Zseq_imp_Zseq)
```
```   252     fix n::nat
```
```   253     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   254       by (rule norm_le)
```
```   255     also have "\<dots> \<le> norm (X n) * B * K"
```
```   256       by (intro mult_mono' order_refl norm_Y norm_ge_zero
```
```   257                 mult_nonneg_nonneg K)
```
```   258     also have "\<dots> = norm (X n) * (B * K)"
```
```   259       by (rule mult_assoc)
```
```   260     finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
```
```   261   qed
```
```   262 qed
```
```   263
```
```   264 lemma (in bounded_bilinear) Bseq_prod_Zseq:
```
```   265   assumes X: "Bseq X"
```
```   266   assumes Y: "Zseq Y"
```
```   267   shows "Zseq (\<lambda>n. X n ** Y n)"
```
```   268 proof -
```
```   269   obtain K where K: "0 \<le> K"
```
```   270     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   271     using nonneg_bounded by fast
```
```   272   obtain B where B: "0 < B"
```
```   273     and norm_X: "\<And>n. norm (X n) \<le> B"
```
```   274     using X [unfolded Bseq_def] by fast
```
```   275   from Y show ?thesis
```
```   276   proof (rule Zseq_imp_Zseq)
```
```   277     fix n::nat
```
```   278     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
```
```   279       by (rule norm_le)
```
```   280     also have "\<dots> \<le> B * norm (Y n) * K"
```
```   281       by (intro mult_mono' order_refl norm_X norm_ge_zero
```
```   282                 mult_nonneg_nonneg K)
```
```   283     also have "\<dots> = norm (Y n) * (B * K)"
```
```   284       by (simp only: mult_ac)
```
```   285     finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
```
```   286   qed
```
```   287 qed
```
```   288
```
```   289 lemma (in bounded_bilinear) Zseq_left:
```
```   290   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
```
```   291 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
```
```   292
```
```   293 lemma (in bounded_bilinear) Zseq_right:
```
```   294   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
```
```   295 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
```
```   296
```
```   297 lemmas Zseq_mult = mult.Zseq
```
```   298 lemmas Zseq_mult_right = mult.Zseq_right
```
```   299 lemmas Zseq_mult_left = mult.Zseq_left
```
```   300
```
```   301
```
```   302 subsection {* Limits of Sequences *}
```
```   303
```
```   304 lemma LIMSEQ_iff:
```
```   305       "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   306 by (rule LIMSEQ_def)
```
```   307
```
```   308 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
```
```   309 by (simp only: LIMSEQ_def Zseq_def)
```
```   310
```
```   311 lemma LIMSEQ_I:
```
```   312   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   313 by (simp add: LIMSEQ_def)
```
```   314
```
```   315 lemma LIMSEQ_D:
```
```   316   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   317 by (simp add: LIMSEQ_def)
```
```   318
```
```   319 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
```
```   320 by (simp add: LIMSEQ_def)
```
```   321
```
```   322 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l = (k = l)"
```
```   323 by (simp add: LIMSEQ_Zseq_iff Zseq_const_iff)
```
```   324
```
```   325 lemma LIMSEQ_norm: "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
```
```   326 apply (simp add: LIMSEQ_def, safe)
```
```   327 apply (drule_tac x="r" in spec, safe)
```
```   328 apply (rule_tac x="no" in exI, safe)
```
```   329 apply (drule_tac x="n" in spec, safe)
```
```   330 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
```
```   331 done
```
```   332
```
```   333 lemma LIMSEQ_ignore_initial_segment:
```
```   334   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   335 apply (rule LIMSEQ_I)
```
```   336 apply (drule (1) LIMSEQ_D)
```
```   337 apply (erule exE, rename_tac N)
```
```   338 apply (rule_tac x=N in exI)
```
```   339 apply simp
```
```   340 done
```
```   341
```
```   342 lemma LIMSEQ_offset:
```
```   343   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   344 apply (rule LIMSEQ_I)
```
```   345 apply (drule (1) LIMSEQ_D)
```
```   346 apply (erule exE, rename_tac N)
```
```   347 apply (rule_tac x="N + k" in exI)
```
```   348 apply clarify
```
```   349 apply (drule_tac x="n - k" in spec)
```
```   350 apply (simp add: le_diff_conv2)
```
```   351 done
```
```   352
```
```   353 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   354 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   355
```
```   356 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   357 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   358
```
```   359 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   360 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   361
```
```   362 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   363   unfolding LIMSEQ_def
```
```   364   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   365
```
```   366
```
```   367 lemma add_diff_add:
```
```   368   fixes a b c d :: "'a::ab_group_add"
```
```   369   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```   370 by simp
```
```   371
```
```   372 lemma minus_diff_minus:
```
```   373   fixes a b :: "'a::ab_group_add"
```
```   374   shows "(- a) - (- b) = - (a - b)"
```
```   375 by simp
```
```   376
```
```   377 lemma LIMSEQ_add: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
```
```   378 by (simp only: LIMSEQ_Zseq_iff add_diff_add Zseq_add)
```
```   379
```
```   380 lemma LIMSEQ_minus: "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
```
```   381 by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
```
```   382
```
```   383 lemma LIMSEQ_minus_cancel: "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
```
```   384 by (drule LIMSEQ_minus, simp)
```
```   385
```
```   386 lemma LIMSEQ_diff: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
```
```   387 by (simp add: diff_minus LIMSEQ_add LIMSEQ_minus)
```
```   388
```
```   389 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   390 by (drule (1) LIMSEQ_diff, simp add: LIMSEQ_const_iff)
```
```   391
```
```   392 lemma (in bounded_linear) LIMSEQ:
```
```   393   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```
```   394 by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
```
```   395
```
```   396 lemma (in bounded_bilinear) LIMSEQ:
```
```   397   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
```
```   398 by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
```
```   399                Zseq_add Zseq Zseq_left Zseq_right)
```
```   400
```
```   401 lemma LIMSEQ_mult:
```
```   402   fixes a b :: "'a::real_normed_algebra"
```
```   403   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
```
```   404 by (rule mult.LIMSEQ)
```
```   405
```
```   406 lemma inverse_diff_inverse:
```
```   407   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   408    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   409 by (simp add: algebra_simps)
```
```   410
```
```   411 lemma Bseq_inverse_lemma:
```
```   412   fixes x :: "'a::real_normed_div_algebra"
```
```   413   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   414 apply (subst nonzero_norm_inverse, clarsimp)
```
```   415 apply (erule (1) le_imp_inverse_le)
```
```   416 done
```
```   417
```
```   418 lemma Bseq_inverse:
```
```   419   fixes a :: "'a::real_normed_div_algebra"
```
```   420   assumes X: "X ----> a"
```
```   421   assumes a: "a \<noteq> 0"
```
```   422   shows "Bseq (\<lambda>n. inverse (X n))"
```
```   423 proof -
```
```   424   from a have "0 < norm a" by simp
```
```   425   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   426   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   427   obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
```
```   428     using LIMSEQ_D [OF X r1] by fast
```
```   429   show ?thesis
```
```   430   proof (rule BseqI2' [rule_format])
```
```   431     fix n assume n: "N \<le> n"
```
```   432     hence 1: "norm (X n - a) < r" by (rule N)
```
```   433     hence 2: "X n \<noteq> 0" using r2 by auto
```
```   434     hence "norm (inverse (X n)) = inverse (norm (X n))"
```
```   435       by (rule nonzero_norm_inverse)
```
```   436     also have "\<dots> \<le> inverse (norm a - r)"
```
```   437     proof (rule le_imp_inverse_le)
```
```   438       show "0 < norm a - r" using r2 by simp
```
```   439     next
```
```   440       have "norm a - norm (X n) \<le> norm (a - X n)"
```
```   441         by (rule norm_triangle_ineq2)
```
```   442       also have "\<dots> = norm (X n - a)"
```
```   443         by (rule norm_minus_commute)
```
```   444       also have "\<dots> < r" using 1 .
```
```   445       finally show "norm a - r \<le> norm (X n)" by simp
```
```   446     qed
```
```   447     finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
```
```   448   qed
```
```   449 qed
```
```   450
```
```   451 lemma LIMSEQ_inverse_lemma:
```
```   452   fixes a :: "'a::real_normed_div_algebra"
```
```   453   shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
```
```   454          \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
```
```   455 apply (subst LIMSEQ_Zseq_iff)
```
```   456 apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
```
```   457 apply (rule Zseq_minus)
```
```   458 apply (rule Zseq_mult_left)
```
```   459 apply (rule mult.Bseq_prod_Zseq)
```
```   460 apply (erule (1) Bseq_inverse)
```
```   461 apply (simp add: LIMSEQ_Zseq_iff)
```
```   462 done
```
```   463
```
```   464 lemma LIMSEQ_inverse:
```
```   465   fixes a :: "'a::real_normed_div_algebra"
```
```   466   assumes X: "X ----> a"
```
```   467   assumes a: "a \<noteq> 0"
```
```   468   shows "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   469 proof -
```
```   470   from a have "0 < norm a" by simp
```
```   471   then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
```
```   472     using LIMSEQ_D [OF X] by fast
```
```   473   hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
```
```   474   hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
```
```   475
```
```   476   from X have "(\<lambda>n. X (n + k)) ----> a"
```
```   477     by (rule LIMSEQ_ignore_initial_segment)
```
```   478   hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
```
```   479     using a k by (rule LIMSEQ_inverse_lemma)
```
```   480   thus "(\<lambda>n. inverse (X n)) ----> inverse a"
```
```   481     by (rule LIMSEQ_offset)
```
```   482 qed
```
```   483
```
```   484 lemma LIMSEQ_divide:
```
```   485   fixes a b :: "'a::real_normed_field"
```
```   486   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
```
```   487 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
```
```   488
```
```   489 lemma LIMSEQ_pow:
```
```   490   fixes a :: "'a::{real_normed_algebra,recpower}"
```
```   491   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
```
```   492 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
```
```   493
```
```   494 lemma LIMSEQ_setsum:
```
```   495   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   496   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
```
```   497 proof (cases "finite S")
```
```   498   case True
```
```   499   thus ?thesis using n
```
```   500   proof (induct)
```
```   501     case empty
```
```   502     show ?case
```
```   503       by (simp add: LIMSEQ_const)
```
```   504   next
```
```   505     case insert
```
```   506     thus ?case
```
```   507       by (simp add: LIMSEQ_add)
```
```   508   qed
```
```   509 next
```
```   510   case False
```
```   511   thus ?thesis
```
```   512     by (simp add: LIMSEQ_const)
```
```   513 qed
```
```   514
```
```   515 lemma LIMSEQ_setprod:
```
```   516   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
```
```   517   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
```
```   518   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
```
```   519 proof (cases "finite S")
```
```   520   case True
```
```   521   thus ?thesis using n
```
```   522   proof (induct)
```
```   523     case empty
```
```   524     show ?case
```
```   525       by (simp add: LIMSEQ_const)
```
```   526   next
```
```   527     case insert
```
```   528     thus ?case
```
```   529       by (simp add: LIMSEQ_mult)
```
```   530   qed
```
```   531 next
```
```   532   case False
```
```   533   thus ?thesis
```
```   534     by (simp add: setprod_def LIMSEQ_const)
```
```   535 qed
```
```   536
```
```   537 lemma LIMSEQ_add_const: "f ----> a ==> (%n.(f n + b)) ----> a + b"
```
```   538 by (simp add: LIMSEQ_add LIMSEQ_const)
```
```   539
```
```   540 (* FIXME: delete *)
```
```   541 lemma LIMSEQ_add_minus:
```
```   542      "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
```
```   543 by (simp only: LIMSEQ_add LIMSEQ_minus)
```
```   544
```
```   545 lemma LIMSEQ_diff_const: "f ----> a ==> (%n.(f n  - b)) ----> a - b"
```
```   546 by (simp add: LIMSEQ_diff LIMSEQ_const)
```
```   547
```
```   548 lemma LIMSEQ_diff_approach_zero:
```
```   549   "g ----> L ==> (%x. f x - g x) ----> 0  ==>
```
```   550      f ----> L"
```
```   551   apply (drule LIMSEQ_add)
```
```   552   apply assumption
```
```   553   apply simp
```
```   554 done
```
```   555
```
```   556 lemma LIMSEQ_diff_approach_zero2:
```
```   557   "f ----> L ==> (%x. f x - g x) ----> 0  ==>
```
```   558      g ----> L";
```
```   559   apply (drule LIMSEQ_diff)
```
```   560   apply assumption
```
```   561   apply simp
```
```   562 done
```
```   563
```
```   564 text{*A sequence tends to zero iff its abs does*}
```
```   565 lemma LIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----> 0) = (X ----> 0)"
```
```   566 by (simp add: LIMSEQ_def)
```
```   567
```
```   568 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
```
```   569 by (simp add: LIMSEQ_def)
```
```   570
```
```   571 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
```
```   572 by (drule LIMSEQ_norm, simp)
```
```   573
```
```   574 text{*An unbounded sequence's inverse tends to 0*}
```
```   575
```
```   576 lemma LIMSEQ_inverse_zero:
```
```   577   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   578 apply (rule LIMSEQ_I)
```
```   579 apply (drule_tac x="inverse r" in spec, safe)
```
```   580 apply (rule_tac x="N" in exI, safe)
```
```   581 apply (drule_tac x="n" in spec, safe)
```
```   582 apply (frule positive_imp_inverse_positive)
```
```   583 apply (frule (1) less_imp_inverse_less)
```
```   584 apply (subgoal_tac "0 < X n", simp)
```
```   585 apply (erule (1) order_less_trans)
```
```   586 done
```
```   587
```
```   588 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   589
```
```   590 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   591 apply (rule LIMSEQ_inverse_zero, safe)
```
```   592 apply (cut_tac x = r in reals_Archimedean2)
```
```   593 apply (safe, rule_tac x = n in exI)
```
```   594 apply (auto simp add: real_of_nat_Suc)
```
```   595 done
```
```   596
```
```   597 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   598 infinity is now easily proved*}
```
```   599
```
```   600 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   601      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   602 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   603
```
```   604 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   605      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   606 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
```
```   607
```
```   608 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   609      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   610 by (cut_tac b=1 in
```
```   611         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
```
```   612
```
```   613 lemma LIMSEQ_le_const:
```
```   614   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   615 apply (rule ccontr, simp only: linorder_not_le)
```
```   616 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
```
```   617 apply clarsimp
```
```   618 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
```
```   619 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
```
```   620 apply simp
```
```   621 done
```
```   622
```
```   623 lemma LIMSEQ_le_const2:
```
```   624   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   625 apply (subgoal_tac "- a \<le> - x", simp)
```
```   626 apply (rule LIMSEQ_le_const)
```
```   627 apply (erule LIMSEQ_minus)
```
```   628 apply simp
```
```   629 done
```
```   630
```
```   631 lemma LIMSEQ_le:
```
```   632   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
```
```   633 apply (subgoal_tac "0 \<le> y - x", simp)
```
```   634 apply (rule LIMSEQ_le_const)
```
```   635 apply (erule (1) LIMSEQ_diff)
```
```   636 apply (simp add: le_diff_eq)
```
```   637 done
```
```   638
```
```   639
```
```   640 subsection {* Convergence *}
```
```   641
```
```   642 lemma limI: "X ----> L ==> lim X = L"
```
```   643 apply (simp add: lim_def)
```
```   644 apply (blast intro: LIMSEQ_unique)
```
```   645 done
```
```   646
```
```   647 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   648 by (simp add: convergent_def)
```
```   649
```
```   650 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   651 by (auto simp add: convergent_def)
```
```   652
```
```   653 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   654 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   655
```
```   656 lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
```
```   657 apply (simp add: convergent_def)
```
```   658 apply (auto dest: LIMSEQ_minus)
```
```   659 apply (drule LIMSEQ_minus, auto)
```
```   660 done
```
```   661
```
```   662 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   663
```
```   664 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   665   unfolding Ex1_def
```
```   666   apply (rule_tac x="nat_rec e f" in exI)
```
```   667   apply (rule conjI)+
```
```   668 apply (rule def_nat_rec_0, simp)
```
```   669 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   670 apply (rule allI, rule impI, rule ext)
```
```   671 apply (erule conjE)
```
```   672 apply (induct_tac x)
```
```   673 apply (simp add: nat_rec_0)
```
```   674 apply (erule_tac x="n" in allE)
```
```   675 apply (simp)
```
```   676 done
```
```   677
```
```   678 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   679
```
```   680 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   681 apply (simp add: subseq_def)
```
```   682 apply (auto dest!: less_imp_Suc_add)
```
```   683 apply (induct_tac k)
```
```   684 apply (auto intro: less_trans)
```
```   685 done
```
```   686
```
```   687 lemma monoseq_Suc:
```
```   688    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
```
```   689                  | (\<forall>n. X (Suc n) \<le> X n))"
```
```   690 apply (simp add: monoseq_def)
```
```   691 apply (auto dest!: le_imp_less_or_eq)
```
```   692 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
```
```   693 apply (induct_tac "ka")
```
```   694 apply (auto intro: order_trans)
```
```   695 apply (erule contrapos_np)
```
```   696 apply (induct_tac "k")
```
```   697 apply (auto intro: order_trans)
```
```   698 done
```
```   699
```
```   700 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```   701 by (simp add: monoseq_def)
```
```   702
```
```   703 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```   704 by (simp add: monoseq_def)
```
```   705
```
```   706 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```   707 by (simp add: monoseq_Suc)
```
```   708
```
```   709 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```   710 by (simp add: monoseq_Suc)
```
```   711
```
```   712 lemma monoseq_minus: assumes "monoseq a"
```
```   713   shows "monoseq (\<lambda> n. - a n)"
```
```   714 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   715   case True
```
```   716   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   717   thus ?thesis by (rule monoI2)
```
```   718 next
```
```   719   case False
```
```   720   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   721   thus ?thesis by (rule monoI1)
```
```   722 qed
```
```   723
```
```   724 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
```
```   725   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
```
```   726          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   727 proof -
```
```   728   { fix x n fix a :: "nat \<Rightarrow> real"
```
```   729     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
```
```   730     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
```
```   731     have "a n \<le> x"
```
```   732     proof (rule ccontr)
```
```   733       assume "\<not> a n \<le> x" hence "x < a n" by auto
```
```   734       hence "0 < a n - x" by auto
```
```   735       from `a ----> x`[THEN LIMSEQ_D, OF this]
```
```   736       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
```
```   737       hence "norm (a (max no n) - x) < a n - x" by auto
```
```   738       moreover
```
```   739       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
```
```   740       hence "x < a (max no n)" by auto
```
```   741       ultimately
```
```   742       have "a (max no n) < a n" by auto
```
```   743       with monotone[where m=n and n="max no n"]
```
```   744       show False by auto
```
```   745     qed
```
```   746   } note top_down = this
```
```   747   { fix x n m fix a :: "nat \<Rightarrow> real"
```
```   748     assume "a ----> x" and "monoseq a" and "a m < x"
```
```   749     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
```
```   750     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   751       case True with top_down and `a ----> x` show ?thesis by auto
```
```   752     next
```
```   753       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
```
```   754       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
```
```   755       hence False using `a m < x` by auto
```
```   756       thus ?thesis ..
```
```   757     qed
```
```   758   } note when_decided = this
```
```   759
```
```   760   show ?thesis
```
```   761   proof (cases "\<exists> m. a m \<noteq> x")
```
```   762     case True then obtain m where "a m \<noteq> x" by auto
```
```   763     show ?thesis
```
```   764     proof (cases "a m < x")
```
```   765       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
```
```   766       show ?thesis by blast
```
```   767     next
```
```   768       case False hence "- a m < - x" using `a m \<noteq> x` by auto
```
```   769       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
```
```   770       show ?thesis by auto
```
```   771     qed
```
```   772   qed auto
```
```   773 qed
```
```   774
```
```   775 text{* for any sequence, there is a mootonic subsequence *}
```
```   776 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   777 proof-
```
```   778   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   779     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   780     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   781     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   782     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   783       using H apply -
```
```   784       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   785       unfolding order_le_less by blast
```
```   786     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   787     {fix n
```
```   788       have "?P (f (Suc n)) (f n)"
```
```   789 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   790 	using H apply -
```
```   791       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   792       unfolding order_le_less by blast
```
```   793     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   794   note fSuc = this
```
```   795     {fix p q assume pq: "p \<ge> f q"
```
```   796       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   797 	by (cases q, simp_all) }
```
```   798     note pqth = this
```
```   799     {fix q
```
```   800       have "f (Suc q) > f q" apply (induct q)
```
```   801 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   802     note fss = this
```
```   803     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   804     {fix a b
```
```   805       have "f a \<le> f (a + b)"
```
```   806       proof(induct b)
```
```   807 	case 0 thus ?case by simp
```
```   808       next
```
```   809 	case (Suc b)
```
```   810 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   811       qed}
```
```   812     note fmon0 = this
```
```   813     have "monoseq (\<lambda>n. s (f n))"
```
```   814     proof-
```
```   815       {fix n
```
```   816 	have "s (f n) \<ge> s (f (Suc n))"
```
```   817 	proof(cases n)
```
```   818 	  case 0
```
```   819 	  assume n0: "n = 0"
```
```   820 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   821 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   822 	next
```
```   823 	  case (Suc m)
```
```   824 	  assume m: "n = Suc m"
```
```   825 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   826 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   827 	qed}
```
```   828       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   829     qed
```
```   830     with th1 have ?thesis by blast}
```
```   831   moreover
```
```   832   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   833     {fix p assume p: "p \<ge> Suc N"
```
```   834       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   835       have "m \<noteq> p" using m(2) by auto
```
```   836       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   837     note th0 = this
```
```   838     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   839     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   840     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   841       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   842     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   843       using N apply -
```
```   844       apply (erule allE[where x="Suc N"], clarsimp)
```
```   845       apply (rule_tac x="m" in exI)
```
```   846       apply auto
```
```   847       apply (subgoal_tac "Suc N \<noteq> m")
```
```   848       apply simp
```
```   849       apply (rule ccontr, simp)
```
```   850       done
```
```   851     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   852     {fix n
```
```   853       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   854 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   855       proof (induct n)
```
```   856 	case 0 thus ?case
```
```   857 	  using f0 N apply auto
```
```   858 	  apply (erule allE[where x="f 0"], clarsimp)
```
```   859 	  apply (rule_tac x="m" in exI, simp)
```
```   860 	  by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   861       next
```
```   862 	case (Suc n)
```
```   863 	from Suc.hyps have Nfn: "N < f n" by blast
```
```   864 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   865 	with Nfn have mN: "m > N" by arith
```
```   866 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   867
```
```   868 	from key have th0: "f (Suc n) > N" by simp
```
```   869 	from N[rule_format, OF th0]
```
```   870 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   871 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   872 	hence "m' > f (Suc n)" using m'(1) by simp
```
```   873 	with key m'(2) show ?case by auto
```
```   874       qed}
```
```   875     note fSuc = this
```
```   876     {fix n
```
```   877       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
```
```   878       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   879     note thf = this
```
```   880     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   881     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   882       apply -
```
```   883       apply (rule disjI1)
```
```   884       apply auto
```
```   885       apply (rule order_less_imp_le)
```
```   886       apply blast
```
```   887       done
```
```   888     then have ?thesis  using sqf by blast}
```
```   889   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   890 qed
```
```   891
```
```   892 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   893 proof(induct n)
```
```   894   case 0 thus ?case by simp
```
```   895 next
```
```   896   case (Suc n)
```
```   897   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   898   have "n < f (Suc n)" by arith
```
```   899   thus ?case by arith
```
```   900 qed
```
```   901
```
```   902 lemma LIMSEQ_subseq_LIMSEQ:
```
```   903   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```   904 apply (auto simp add: LIMSEQ_def)
```
```   905 apply (drule_tac x=r in spec, clarify)
```
```   906 apply (rule_tac x=no in exI, clarify)
```
```   907 apply (blast intro: seq_suble le_trans dest!: spec)
```
```   908 done
```
```   909
```
```   910 subsection {* Bounded Monotonic Sequences *}
```
```   911
```
```   912
```
```   913 text{*Bounded Sequence*}
```
```   914
```
```   915 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   916 by (simp add: Bseq_def)
```
```   917
```
```   918 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   919 by (auto simp add: Bseq_def)
```
```   920
```
```   921 lemma lemma_NBseq_def:
```
```   922      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
```
```   923       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   924 apply auto
```
```   925  prefer 2 apply force
```
```   926 apply (cut_tac x = K in reals_Archimedean2, clarify)
```
```   927 apply (rule_tac x = n in exI, clarify)
```
```   928 apply (drule_tac x = na in spec)
```
```   929 apply (auto simp add: real_of_nat_Suc)
```
```   930 done
```
```   931
```
```   932 text{* alternative definition for Bseq *}
```
```   933 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   934 apply (simp add: Bseq_def)
```
```   935 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   936 done
```
```   937
```
```   938 lemma lemma_NBseq_def2:
```
```   939      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   940 apply (subst lemma_NBseq_def, auto)
```
```   941 apply (rule_tac x = "Suc N" in exI)
```
```   942 apply (rule_tac [2] x = N in exI)
```
```   943 apply (auto simp add: real_of_nat_Suc)
```
```   944  prefer 2 apply (blast intro: order_less_imp_le)
```
```   945 apply (drule_tac x = n in spec, simp)
```
```   946 done
```
```   947
```
```   948 (* yet another definition for Bseq *)
```
```   949 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   950 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   951
```
```   952 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   953
```
```   954 lemma Bseq_isUb:
```
```   955   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   956 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   957
```
```   958
```
```   959 text{* Use completeness of reals (supremum property)
```
```   960    to show that any bounded sequence has a least upper bound*}
```
```   961
```
```   962 lemma Bseq_isLub:
```
```   963   "!!(X::nat=>real). Bseq X ==>
```
```   964    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   965 by (blast intro: reals_complete Bseq_isUb)
```
```   966
```
```   967 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   968
```
```   969 lemma lemma_converg1:
```
```   970      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
```
```   971                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
```
```   972                |] ==> \<forall>n \<ge> ma. X n = X ma"
```
```   973 apply safe
```
```   974 apply (drule_tac y = "X n" in isLubD2)
```
```   975 apply (blast dest: order_antisym)+
```
```   976 done
```
```   977
```
```   978 text{* The best of both worlds: Easier to prove this result as a standard
```
```   979    theorem and then use equivalence to "transfer" it into the
```
```   980    equivalent nonstandard form if needed!*}
```
```   981
```
```   982 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   983 apply (simp add: LIMSEQ_def)
```
```   984 apply (rule_tac x = "X m" in exI, safe)
```
```   985 apply (rule_tac x = m in exI, safe)
```
```   986 apply (drule spec, erule impE, auto)
```
```   987 done
```
```   988
```
```   989 lemma lemma_converg2:
```
```   990    "!!(X::nat=>real).
```
```   991     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
```
```   992 apply safe
```
```   993 apply (drule_tac y = "X m" in isLubD2)
```
```   994 apply (auto dest!: order_le_imp_less_or_eq)
```
```   995 done
```
```   996
```
```   997 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
```
```   998 by (rule setleI [THEN isUbI], auto)
```
```   999
```
```  1000 text{* FIXME: @{term "U - T < U"} is redundant *}
```
```  1001 lemma lemma_converg4: "!!(X::nat=> real).
```
```  1002                [| \<forall>m. X m ~= U;
```
```  1003                   isLub UNIV {x. \<exists>n. X n = x} U;
```
```  1004                   0 < T;
```
```  1005                   U + - T < U
```
```  1006                |] ==> \<exists>m. U + -T < X m & X m < U"
```
```  1007 apply (drule lemma_converg2, assumption)
```
```  1008 apply (rule ccontr, simp)
```
```  1009 apply (simp add: linorder_not_less)
```
```  1010 apply (drule lemma_converg3)
```
```  1011 apply (drule isLub_le_isUb, assumption)
```
```  1012 apply (auto dest: order_less_le_trans)
```
```  1013 done
```
```  1014
```
```  1015 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```  1016
```
```  1017 lemma Bseq_mono_convergent:
```
```  1018      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
```
```  1019 apply (simp add: convergent_def)
```
```  1020 apply (frule Bseq_isLub, safe)
```
```  1021 apply (case_tac "\<exists>m. X m = U", auto)
```
```  1022 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
```
```  1023 (* second case *)
```
```  1024 apply (rule_tac x = U in exI)
```
```  1025 apply (subst LIMSEQ_iff, safe)
```
```  1026 apply (frule lemma_converg2, assumption)
```
```  1027 apply (drule lemma_converg4, auto)
```
```  1028 apply (rule_tac x = m in exI, safe)
```
```  1029 apply (subgoal_tac "X m \<le> X n")
```
```  1030  prefer 2 apply blast
```
```  1031 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
```
```  1032 done
```
```  1033
```
```  1034 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```  1035 by (simp add: Bseq_def)
```
```  1036
```
```  1037 text{*Main monotonicity theorem*}
```
```  1038 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
```
```  1039 apply (simp add: monoseq_def, safe)
```
```  1040 apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
```
```  1041 apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
```
```  1042 apply (auto intro!: Bseq_mono_convergent)
```
```  1043 done
```
```  1044
```
```  1045 subsubsection{*Increasing and Decreasing Series*}
```
```  1046
```
```  1047 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```  1048   by (simp add: incseq_def monoseq_def)
```
```  1049
```
```  1050 lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
```
```  1051   using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
```
```  1052 proof
```
```  1053   assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
```
```  1054   thus ?thesis by simp
```
```  1055 next
```
```  1056   assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
```
```  1057   hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
```
```  1058     by (auto simp add: incseq_def intro: order_antisym)
```
```  1059   have X: "!!n. X n = X 0"
```
```  1060     by (blast intro: const [of 0])
```
```  1061   have "X = (\<lambda>n. X 0)"
```
```  1062     by (blast intro: ext X)
```
```  1063   hence "L = X 0" using LIMSEQ_const [of "X 0"]
```
```  1064     by (auto intro: LIMSEQ_unique lim)
```
```  1065   thus ?thesis
```
```  1066     by (blast intro: eq_refl X)
```
```  1067 qed
```
```  1068
```
```  1069 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```  1070   by (simp add: decseq_def monoseq_def)
```
```  1071
```
```  1072 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
```
```  1073   by (simp add: decseq_def incseq_def)
```
```  1074
```
```  1075
```
```  1076 lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
```
```  1077 proof -
```
```  1078   have inc: "incseq (\<lambda>n. - X n)" using dec
```
```  1079     by (simp add: decseq_eq_incseq)
```
```  1080   have "- X n \<le> - L"
```
```  1081     by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim)
```
```  1082   thus ?thesis
```
```  1083     by simp
```
```  1084 qed
```
```  1085
```
```  1086 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```  1087
```
```  1088 text{*alternative formulation for boundedness*}
```
```  1089 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```  1090 apply (unfold Bseq_def, safe)
```
```  1091 apply (rule_tac [2] x = "k + norm x" in exI)
```
```  1092 apply (rule_tac x = K in exI, simp)
```
```  1093 apply (rule exI [where x = 0], auto)
```
```  1094 apply (erule order_less_le_trans, simp)
```
```  1095 apply (drule_tac x=n in spec, fold diff_def)
```
```  1096 apply (drule order_trans [OF norm_triangle_ineq2])
```
```  1097 apply simp
```
```  1098 done
```
```  1099
```
```  1100 text{*alternative formulation for boundedness*}
```
```  1101 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```  1102 apply safe
```
```  1103 apply (simp add: Bseq_def, safe)
```
```  1104 apply (rule_tac x = "K + norm (X N)" in exI)
```
```  1105 apply auto
```
```  1106 apply (erule order_less_le_trans, simp)
```
```  1107 apply (rule_tac x = N in exI, safe)
```
```  1108 apply (drule_tac x = n in spec)
```
```  1109 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```  1110 apply (auto simp add: Bseq_iff2)
```
```  1111 done
```
```  1112
```
```  1113 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```  1114 apply (simp add: Bseq_def)
```
```  1115 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```  1116 apply (drule_tac x = n in spec, arith)
```
```  1117 done
```
```  1118
```
```  1119
```
```  1120 subsection {* Cauchy Sequences *}
```
```  1121
```
```  1122 lemma CauchyI:
```
```  1123   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
```
```  1124 by (simp add: Cauchy_def)
```
```  1125
```
```  1126 lemma CauchyD:
```
```  1127   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1128 by (simp add: Cauchy_def)
```
```  1129
```
```  1130 lemma Cauchy_subseq_Cauchy:
```
```  1131   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```  1132 apply (auto simp add: Cauchy_def)
```
```  1133 apply (drule_tac x=e in spec, clarify)
```
```  1134 apply (rule_tac x=M in exI, clarify)
```
```  1135 apply (blast intro: seq_suble le_trans dest!: spec)
```
```  1136 done
```
```  1137
```
```  1138 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1139
```
```  1140 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1141   proof mechanization rather than the nonstandard proof*}
```
```  1142
```
```  1143 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1144           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1145 apply (clarify, drule spec, drule (1) mp)
```
```  1146 apply (simp only: norm_minus_commute)
```
```  1147 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1148 apply simp
```
```  1149 done
```
```  1150
```
```  1151 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```  1152 apply (simp add: Cauchy_def)
```
```  1153 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```  1154 apply (drule_tac x="M" in spec, simp)
```
```  1155 apply (drule lemmaCauchy)
```
```  1156 apply (rule_tac k="M" in Bseq_offset)
```
```  1157 apply (simp add: Bseq_def)
```
```  1158 apply (rule_tac x="1 + norm (X M)" in exI)
```
```  1159 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```  1160 apply (simp add: order_less_imp_le)
```
```  1161 done
```
```  1162
```
```  1163 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1164
```
```  1165 axclass banach \<subseteq> real_normed_vector
```
```  1166   Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1167
```
```  1168 theorem LIMSEQ_imp_Cauchy:
```
```  1169   assumes X: "X ----> a" shows "Cauchy X"
```
```  1170 proof (rule CauchyI)
```
```  1171   fix e::real assume "0 < e"
```
```  1172   hence "0 < e/2" by simp
```
```  1173   with X have "\<exists>N. \<forall>n\<ge>N. norm (X n - a) < e/2" by (rule LIMSEQ_D)
```
```  1174   then obtain N where N: "\<forall>n\<ge>N. norm (X n - a) < e/2" ..
```
```  1175   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < e"
```
```  1176   proof (intro exI allI impI)
```
```  1177     fix m assume "N \<le> m"
```
```  1178     hence m: "norm (X m - a) < e/2" using N by fast
```
```  1179     fix n assume "N \<le> n"
```
```  1180     hence n: "norm (X n - a) < e/2" using N by fast
```
```  1181     have "norm (X m - X n) = norm ((X m - a) - (X n - a))" by simp
```
```  1182     also have "\<dots> \<le> norm (X m - a) + norm (X n - a)"
```
```  1183       by (rule norm_triangle_ineq4)
```
```  1184     also from m n have "\<dots> < e" by(simp add:field_simps)
```
```  1185     finally show "norm (X m - X n) < e" .
```
```  1186   qed
```
```  1187 qed
```
```  1188
```
```  1189 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1190 unfolding convergent_def
```
```  1191 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1192
```
```  1193 text {*
```
```  1194 Proof that Cauchy sequences converge based on the one from
```
```  1195 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1196 *}
```
```  1197
```
```  1198 text {*
```
```  1199   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1200   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1201 *}
```
```  1202
```
```  1203 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1204 by (simp add: isUbI setleI)
```
```  1205
```
```  1206 lemma real_abs_diff_less_iff:
```
```  1207   "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
```
```  1208 by auto
```
```  1209
```
```  1210 locale real_Cauchy =
```
```  1211   fixes X :: "nat \<Rightarrow> real"
```
```  1212   assumes X: "Cauchy X"
```
```  1213   fixes S :: "real set"
```
```  1214   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1215
```
```  1216 lemma real_CauchyI:
```
```  1217   assumes "Cauchy X"
```
```  1218   shows "real_Cauchy X"
```
```  1219   proof qed (fact assms)
```
```  1220
```
```  1221 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
```
```  1222 by (unfold S_def, auto)
```
```  1223
```
```  1224 lemma (in real_Cauchy) bound_isUb:
```
```  1225   assumes N: "\<forall>n\<ge>N. X n < x"
```
```  1226   shows "isUb UNIV S x"
```
```  1227 proof (rule isUb_UNIV_I)
```
```  1228   fix y::real assume "y \<in> S"
```
```  1229   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1230     by (simp add: S_def)
```
```  1231   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1232   hence "y < X (max M N)" by simp
```
```  1233   also have "\<dots> < x" using N by simp
```
```  1234   finally show "y \<le> x"
```
```  1235     by (rule order_less_imp_le)
```
```  1236 qed
```
```  1237
```
```  1238 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
```
```  1239 proof (rule reals_complete)
```
```  1240   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```  1241     using CauchyD [OF X zero_less_one] by fast
```
```  1242   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```  1243   show "\<exists>x. x \<in> S"
```
```  1244   proof
```
```  1245     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1246       by (simp add: real_abs_diff_less_iff)
```
```  1247     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1248   qed
```
```  1249   show "\<exists>u. isUb UNIV S u"
```
```  1250   proof
```
```  1251     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1252       by (simp add: real_abs_diff_less_iff)
```
```  1253     thus "isUb UNIV S (X N + 1)"
```
```  1254       by (rule bound_isUb)
```
```  1255   qed
```
```  1256 qed
```
```  1257
```
```  1258 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
```
```  1259   assumes x: "isLub UNIV S x"
```
```  1260   shows "X ----> x"
```
```  1261 proof (rule LIMSEQ_I)
```
```  1262   fix r::real assume "0 < r"
```
```  1263   hence r: "0 < r/2" by simp
```
```  1264   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```  1265     using CauchyD [OF X r] by fast
```
```  1266   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```  1267   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1268     by (simp only: real_norm_def real_abs_diff_less_iff)
```
```  1269
```
```  1270   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1271   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1272   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1273
```
```  1274   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1275   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1276   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1277
```
```  1278   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```  1279   proof (intro exI allI impI)
```
```  1280     fix n assume n: "N \<le> n"
```
```  1281     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1282     thus "norm (X n - x) < r" using 1 2
```
```  1283       by (simp add: real_abs_diff_less_iff)
```
```  1284   qed
```
```  1285 qed
```
```  1286
```
```  1287 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
```
```  1288 proof -
```
```  1289   obtain x where "isLub UNIV S x"
```
```  1290     using isLub_ex by fast
```
```  1291   hence "X ----> x"
```
```  1292     by (rule isLub_imp_LIMSEQ)
```
```  1293   thus ?thesis ..
```
```  1294 qed
```
```  1295
```
```  1296 lemma real_Cauchy_convergent:
```
```  1297   fixes X :: "nat \<Rightarrow> real"
```
```  1298   shows "Cauchy X \<Longrightarrow> convergent X"
```
```  1299 unfolding convergent_def
```
```  1300 by (rule real_Cauchy.LIMSEQ_ex)
```
```  1301  (rule real_CauchyI)
```
```  1302
```
```  1303 instance real :: banach
```
```  1304 by intro_classes (rule real_Cauchy_convergent)
```
```  1305
```
```  1306 lemma Cauchy_convergent_iff:
```
```  1307   fixes X :: "nat \<Rightarrow> 'a::banach"
```
```  1308   shows "Cauchy X = convergent X"
```
```  1309 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1310
```
```  1311 lemma convergent_subseq_convergent:
```
```  1312   fixes X :: "nat \<Rightarrow> 'a::banach"
```
```  1313   shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```  1314   by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
```
```  1315
```
```  1316
```
```  1317 subsection {* Power Sequences *}
```
```  1318
```
```  1319 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1320 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1321   also fact that bounded and monotonic sequence converges.*}
```
```  1322
```
```  1323 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1324 apply (simp add: Bseq_def)
```
```  1325 apply (rule_tac x = 1 in exI)
```
```  1326 apply (simp add: power_abs)
```
```  1327 apply (auto dest: power_mono)
```
```  1328 done
```
```  1329
```
```  1330 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1331 apply (clarify intro!: mono_SucI2)
```
```  1332 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1333 done
```
```  1334
```
```  1335 lemma convergent_realpow:
```
```  1336   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1337 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1338
```
```  1339 lemma LIMSEQ_inverse_realpow_zero_lemma:
```
```  1340   fixes x :: real
```
```  1341   assumes x: "0 \<le> x"
```
```  1342   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1343 apply (induct n)
```
```  1344 apply simp
```
```  1345 apply simp
```
```  1346 apply (rule order_trans)
```
```  1347 prefer 2
```
```  1348 apply (erule mult_left_mono)
```
```  1349 apply (rule add_increasing [OF x], simp)
```
```  1350 apply (simp add: real_of_nat_Suc)
```
```  1351 apply (simp add: ring_distribs)
```
```  1352 apply (simp add: mult_nonneg_nonneg x)
```
```  1353 done
```
```  1354
```
```  1355 lemma LIMSEQ_inverse_realpow_zero:
```
```  1356   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1357 proof (rule LIMSEQ_inverse_zero [rule_format])
```
```  1358   fix y :: real
```
```  1359   assume x: "1 < x"
```
```  1360   hence "0 < x - 1" by simp
```
```  1361   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
```
```  1362     by (rule reals_Archimedean3)
```
```  1363   hence "\<exists>N::nat. y < real N * (x - 1)" ..
```
```  1364   then obtain N::nat where "y < real N * (x - 1)" ..
```
```  1365   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
```
```  1366   also have "\<dots> \<le> (x - 1 + 1) ^ N"
```
```  1367     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
```
```  1368   also have "\<dots> = x ^ N" by simp
```
```  1369   finally have "y < x ^ N" .
```
```  1370   hence "\<forall>n\<ge>N. y < x ^ n"
```
```  1371     apply clarify
```
```  1372     apply (erule order_less_le_trans)
```
```  1373     apply (erule power_increasing)
```
```  1374     apply (rule order_less_imp_le [OF x])
```
```  1375     done
```
```  1376   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
```
```  1377 qed
```
```  1378
```
```  1379 lemma LIMSEQ_realpow_zero:
```
```  1380   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1381 proof (cases)
```
```  1382   assume "x = 0"
```
```  1383   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
```
```  1384   thus ?thesis by (rule LIMSEQ_imp_Suc)
```
```  1385 next
```
```  1386   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1387   hence x0: "0 < x" by simp
```
```  1388   assume x1: "x < 1"
```
```  1389   from x0 x1 have "1 < inverse x"
```
```  1390     by (rule real_inverse_gt_one)
```
```  1391   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1392     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1393   thus ?thesis by (simp add: power_inverse)
```
```  1394 qed
```
```  1395
```
```  1396 lemma LIMSEQ_power_zero:
```
```  1397   fixes x :: "'a::{real_normed_algebra_1,recpower}"
```
```  1398   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1399 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1400 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
```
```  1401 apply (simp add: power_abs norm_power_ineq)
```
```  1402 done
```
```  1403
```
```  1404 lemma LIMSEQ_divide_realpow_zero:
```
```  1405   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
```
```  1406 apply (cut_tac a = a and x1 = "inverse x" in
```
```  1407         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
```
```  1408 apply (auto simp add: divide_inverse power_inverse)
```
```  1409 apply (simp add: inverse_eq_divide pos_divide_less_eq)
```
```  1410 done
```
```  1411
```
```  1412 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1413
```
```  1414 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
```
```  1415 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1416
```
```  1417 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
```
```  1418 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
```
```  1419 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
```
```  1420 done
```
```  1421
```
```  1422 end
```