src/HOL/SEQ.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 50937 d249ef928ae1 child 51328 d63ec23c9125 permissions -rw-r--r--
introduce order topology
```     1 (*  Title:      HOL/SEQ.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Jeremy Avigad
```
```     5     Author:     Brian Huffman
```
```     6
```
```     7 Convergence of sequences and series.
```
```     8 *)
```
```     9
```
```    10 header {* Sequences and Convergence *}
```
```    11
```
```    12 theory SEQ
```
```    13 imports Limits RComplete
```
```    14 begin
```
```    15
```
```    16 subsection {* Monotone sequences and subsequences *}
```
```    17
```
```    18 definition
```
```    19   monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```    20     --{*Definition of monotonicity.
```
```    21         The use of disjunction here complicates proofs considerably.
```
```    22         One alternative is to add a Boolean argument to indicate the direction.
```
```    23         Another is to develop the notions of increasing and decreasing first.*}
```
```    24   "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
```
```    25
```
```    26 definition
```
```    27   incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```    28     --{*Increasing sequence*}
```
```    29   "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
```
```    30
```
```    31 definition
```
```    32   decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```    33     --{*Decreasing sequence*}
```
```    34   "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```    35
```
```    36 definition
```
```    37   subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
```
```    38     --{*Definition of subsequence*}
```
```    39   "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
```
```    40
```
```    41 lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
```
```    42   unfolding mono_def incseq_def by auto
```
```    43
```
```    44 lemma incseq_SucI:
```
```    45   "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
```
```    46   using lift_Suc_mono_le[of X]
```
```    47   by (auto simp: incseq_def)
```
```    48
```
```    49 lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
```
```    50   by (auto simp: incseq_def)
```
```    51
```
```    52 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
```
```    53   using incseqD[of A i "Suc i"] by auto
```
```    54
```
```    55 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
```
```    56   by (auto intro: incseq_SucI dest: incseq_SucD)
```
```    57
```
```    58 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
```
```    59   unfolding incseq_def by auto
```
```    60
```
```    61 lemma decseq_SucI:
```
```    62   "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
```
```    63   using order.lift_Suc_mono_le[OF dual_order, of X]
```
```    64   by (auto simp: decseq_def)
```
```    65
```
```    66 lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
```
```    67   by (auto simp: decseq_def)
```
```    68
```
```    69 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
```
```    70   using decseqD[of A i "Suc i"] by auto
```
```    71
```
```    72 lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
```
```    73   by (auto intro: decseq_SucI dest: decseq_SucD)
```
```    74
```
```    75 lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
```
```    76   unfolding decseq_def by auto
```
```    77
```
```    78 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
```
```    79   unfolding monoseq_def incseq_def decseq_def ..
```
```    80
```
```    81 lemma monoseq_Suc:
```
```    82   "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
```
```    83   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
```
```    84
```
```    85 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```    86 by (simp add: monoseq_def)
```
```    87
```
```    88 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```    89 by (simp add: monoseq_def)
```
```    90
```
```    91 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```    92 by (simp add: monoseq_Suc)
```
```    93
```
```    94 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```    95 by (simp add: monoseq_Suc)
```
```    96
```
```    97 lemma monoseq_minus:
```
```    98   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
```
```    99   assumes "monoseq a"
```
```   100   shows "monoseq (\<lambda> n. - a n)"
```
```   101 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```   102   case True
```
```   103   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```   104   thus ?thesis by (rule monoI2)
```
```   105 next
```
```   106   case False
```
```   107   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```   108   thus ?thesis by (rule monoI1)
```
```   109 qed
```
```   110
```
```   111 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```   112
```
```   113 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```   114 apply (simp add: subseq_def)
```
```   115 apply (auto dest!: less_imp_Suc_add)
```
```   116 apply (induct_tac k)
```
```   117 apply (auto intro: less_trans)
```
```   118 done
```
```   119
```
```   120 text{* for any sequence, there is a monotonic subsequence *}
```
```   121 lemma seq_monosub:
```
```   122   fixes s :: "nat => 'a::linorder"
```
```   123   shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   124 proof cases
```
```   125   let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
```
```   126   assume *: "\<forall>n. \<exists>p. ?P p n"
```
```   127   def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
```
```   128   have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
```
```   129   have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
```
```   130   have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
```
```   131   have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
```
```   132   then have "subseq f" unfolding subseq_Suc_iff by auto
```
```   133   moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
```
```   134   proof (intro disjI2 allI)
```
```   135     fix n show "s (f (Suc n)) \<le> s (f n)"
```
```   136     proof (cases n)
```
```   137       case 0 with P_Suc[of 0] P_0 show ?thesis by auto
```
```   138     next
```
```   139       case (Suc m)
```
```   140       from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   141       with P_Suc Suc show ?thesis by simp
```
```   142     qed
```
```   143   qed
```
```   144   ultimately show ?thesis by auto
```
```   145 next
```
```   146   let "?P p m" = "m < p \<and> s m < s p"
```
```   147   assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
```
```   148   then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
```
```   149   def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
```
```   150   have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
```
```   151   have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
```
```   152   have P_0: "?P (f 0) (Suc N)"
```
```   153     unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
```
```   154   { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
```
```   155       unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
```
```   156   note P' = this
```
```   157   { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
```
```   158       by (induct i) (insert P_0 P', auto) }
```
```   159   then have "subseq f" "monoseq (\<lambda>x. s (f x))"
```
```   160     unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
```
```   161   then show ?thesis by auto
```
```   162 qed
```
```   163
```
```   164 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   165 proof(induct n)
```
```   166   case 0 thus ?case by simp
```
```   167 next
```
```   168   case (Suc n)
```
```   169   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   170   have "n < f (Suc n)" by arith
```
```   171   thus ?case by arith
```
```   172 qed
```
```   173
```
```   174 lemma eventually_subseq:
```
```   175   "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
```
```   176   unfolding eventually_sequentially by (metis seq_suble le_trans)
```
```   177
```
```   178 lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
```
```   179   unfolding filterlim_iff by (metis eventually_subseq)
```
```   180
```
```   181 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
```
```   182   unfolding subseq_def by simp
```
```   183
```
```   184 lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
```
```   185   using assms by (auto simp: subseq_def)
```
```   186
```
```   187 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```   188   by (simp add: incseq_def monoseq_def)
```
```   189
```
```   190 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```   191   by (simp add: decseq_def monoseq_def)
```
```   192
```
```   193 lemma decseq_eq_incseq:
```
```   194   fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
```
```   195   by (simp add: decseq_def incseq_def)
```
```   196
```
```   197 lemma INT_decseq_offset:
```
```   198   assumes "decseq F"
```
```   199   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
```
```   200 proof safe
```
```   201   fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
```
```   202   show "x \<in> F i"
```
```   203   proof cases
```
```   204     from x have "x \<in> F n" by auto
```
```   205     also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
```
```   206       unfolding decseq_def by simp
```
```   207     finally show ?thesis .
```
```   208   qed (insert x, simp)
```
```   209 qed auto
```
```   210
```
```   211 subsection {* Defintions of limits *}
```
```   212
```
```   213 abbreviation (in topological_space)
```
```   214   LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```   215     ("((_)/ ----> (_))" [60, 60] 60) where
```
```   216   "X ----> L \<equiv> (X ---> L) sequentially"
```
```   217
```
```   218 definition
```
```   219   lim :: "(nat \<Rightarrow> 'a::t2_space) \<Rightarrow> 'a" where
```
```   220     --{*Standard definition of limit using choice operator*}
```
```   221   "lim X = (THE L. X ----> L)"
```
```   222
```
```   223 definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```   224   "convergent X = (\<exists>L. X ----> L)"
```
```   225
```
```   226 definition
```
```   227   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
```
```   228     --{*Standard definition for bounded sequence*}
```
```   229   "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
```
```   230
```
```   231 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```   232   "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```   233
```
```   234
```
```   235 subsection {* Bounded Sequences *}
```
```   236
```
```   237 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
```
```   238 unfolding Bseq_def
```
```   239 proof (intro exI conjI allI)
```
```   240   show "0 < max K 1" by simp
```
```   241 next
```
```   242   fix n::nat
```
```   243   have "norm (X n) \<le> K" by (rule K)
```
```   244   thus "norm (X n) \<le> max K 1" by simp
```
```   245 qed
```
```   246
```
```   247 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   248 unfolding Bseq_def by auto
```
```   249
```
```   250 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
```
```   251 proof (rule BseqI')
```
```   252   let ?A = "norm ` X ` {..N}"
```
```   253   have 1: "finite ?A" by simp
```
```   254   fix n::nat
```
```   255   show "norm (X n) \<le> max K (Max ?A)"
```
```   256   proof (cases rule: linorder_le_cases)
```
```   257     assume "n \<ge> N"
```
```   258     hence "norm (X n) \<le> K" using K by simp
```
```   259     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   260   next
```
```   261     assume "n \<le> N"
```
```   262     hence "norm (X n) \<in> ?A" by simp
```
```   263     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
```
```   264     thus "norm (X n) \<le> max K (Max ?A)" by simp
```
```   265   qed
```
```   266 qed
```
```   267
```
```   268 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```   269 unfolding Bseq_def by auto
```
```   270
```
```   271 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```   272 apply (erule BseqE)
```
```   273 apply (rule_tac N="k" and K="K" in BseqI2')
```
```   274 apply clarify
```
```   275 apply (drule_tac x="n - k" in spec, simp)
```
```   276 done
```
```   277
```
```   278 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
```
```   279 unfolding Bfun_def eventually_sequentially
```
```   280 apply (rule iffI)
```
```   281 apply (simp add: Bseq_def)
```
```   282 apply (auto intro: BseqI2')
```
```   283 done
```
```   284
```
```   285
```
```   286 subsection {* Limits of Sequences *}
```
```   287
```
```   288 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
```
```   289   by simp
```
```   290
```
```   291 lemma LIMSEQ_def: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```   292 unfolding tendsto_iff eventually_sequentially ..
```
```   293
```
```   294 lemma LIMSEQ_iff:
```
```   295   fixes L :: "'a::real_normed_vector"
```
```   296   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```   297 unfolding LIMSEQ_def dist_norm ..
```
```   298
```
```   299 lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```   300   unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
```
```   301
```
```   302 lemma metric_LIMSEQ_I:
```
```   303   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
```
```   304 by (simp add: LIMSEQ_def)
```
```   305
```
```   306 lemma metric_LIMSEQ_D:
```
```   307   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```   308 by (simp add: LIMSEQ_def)
```
```   309
```
```   310 lemma LIMSEQ_I:
```
```   311   fixes L :: "'a::real_normed_vector"
```
```   312   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```   313 by (simp add: LIMSEQ_iff)
```
```   314
```
```   315 lemma LIMSEQ_D:
```
```   316   fixes L :: "'a::real_normed_vector"
```
```   317   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```   318 by (simp add: LIMSEQ_iff)
```
```   319
```
```   320 lemma LIMSEQ_const_iff:
```
```   321   fixes k l :: "'a::t2_space"
```
```   322   shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
```
```   323   using trivial_limit_sequentially by (rule tendsto_const_iff)
```
```   324
```
```   325 lemma LIMSEQ_ignore_initial_segment:
```
```   326   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```   327 apply (rule topological_tendstoI)
```
```   328 apply (drule (2) topological_tendstoD)
```
```   329 apply (simp only: eventually_sequentially)
```
```   330 apply (erule exE, rename_tac N)
```
```   331 apply (rule_tac x=N in exI)
```
```   332 apply simp
```
```   333 done
```
```   334
```
```   335 lemma LIMSEQ_offset:
```
```   336   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```   337 apply (rule topological_tendstoI)
```
```   338 apply (drule (2) topological_tendstoD)
```
```   339 apply (simp only: eventually_sequentially)
```
```   340 apply (erule exE, rename_tac N)
```
```   341 apply (rule_tac x="N + k" in exI)
```
```   342 apply clarify
```
```   343 apply (drule_tac x="n - k" in spec)
```
```   344 apply (simp add: le_diff_conv2)
```
```   345 done
```
```   346
```
```   347 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```   348 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```   349
```
```   350 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```   351 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```   352
```
```   353 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```   354 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```   355
```
```   356 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```   357   unfolding tendsto_def eventually_sequentially
```
```   358   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```   359
```
```   360 lemma LIMSEQ_unique:
```
```   361   fixes a b :: "'a::t2_space"
```
```   362   shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```   363   using trivial_limit_sequentially by (rule tendsto_unique)
```
```   364
```
```   365 lemma increasing_LIMSEQ:
```
```   366   fixes f :: "nat \<Rightarrow> real"
```
```   367   assumes inc: "\<And>n. f n \<le> f (Suc n)"
```
```   368       and bdd: "\<And>n. f n \<le> l"
```
```   369       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```   370   shows "f ----> l"
```
```   371 proof (rule increasing_tendsto)
```
```   372   fix x assume "x < l"
```
```   373   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
```
```   374     by auto
```
```   375   from en[OF `0 < e`] obtain n where "l - e \<le> f n"
```
```   376     by (auto simp: field_simps)
```
```   377   with `e < l - x` `0 < e` have "x < f n" by simp
```
```   378   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
```
```   379     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
```
```   380 qed (insert bdd, auto)
```
```   381
```
```   382 lemma Bseq_inverse_lemma:
```
```   383   fixes x :: "'a::real_normed_div_algebra"
```
```   384   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   385 apply (subst nonzero_norm_inverse, clarsimp)
```
```   386 apply (erule (1) le_imp_inverse_le)
```
```   387 done
```
```   388
```
```   389 lemma Bseq_inverse:
```
```   390   fixes a :: "'a::real_normed_div_algebra"
```
```   391   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```   392 unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
```
```   393
```
```   394 lemma LIMSEQ_diff_approach_zero:
```
```   395   fixes L :: "'a::real_normed_vector"
```
```   396   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```   397   by (drule (1) tendsto_add, simp)
```
```   398
```
```   399 lemma LIMSEQ_diff_approach_zero2:
```
```   400   fixes L :: "'a::real_normed_vector"
```
```   401   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
```
```   402   by (drule (1) tendsto_diff, simp)
```
```   403
```
```   404 text{*An unbounded sequence's inverse tends to 0*}
```
```   405
```
```   406 lemma LIMSEQ_inverse_zero:
```
```   407   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```   408   apply (rule filterlim_compose[OF tendsto_inverse_0])
```
```   409   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
```
```   410   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
```
```   411   done
```
```   412
```
```   413 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```   414
```
```   415 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```   416   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
```
```   417             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
```
```   418
```
```   419 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```   420 infinity is now easily proved*}
```
```   421
```
```   422 lemma LIMSEQ_inverse_real_of_nat_add:
```
```   423      "(%n. r + inverse(real(Suc n))) ----> r"
```
```   424   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
```
```   425
```
```   426 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```   427      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```   428   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
```
```   429   by auto
```
```   430
```
```   431 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```   432      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```   433   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
```
```   434   by auto
```
```   435
```
```   436 lemma LIMSEQ_le_const:
```
```   437   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```   438   using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
```
```   439
```
```   440 lemma LIMSEQ_le:
```
```   441   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
```
```   442   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
```
```   443
```
```   444 lemma LIMSEQ_le_const2:
```
```   445   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```   446   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
```
```   447
```
```   448 subsection {* Convergence *}
```
```   449
```
```   450 lemma limI: "X ----> L ==> lim X = L"
```
```   451 apply (simp add: lim_def)
```
```   452 apply (blast intro: LIMSEQ_unique)
```
```   453 done
```
```   454
```
```   455 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```   456 by (simp add: convergent_def)
```
```   457
```
```   458 lemma convergentI: "(X ----> L) ==> convergent X"
```
```   459 by (auto simp add: convergent_def)
```
```   460
```
```   461 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```   462 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```   463
```
```   464 lemma convergent_const: "convergent (\<lambda>n. c)"
```
```   465   by (rule convergentI, rule tendsto_const)
```
```   466
```
```   467 lemma convergent_add:
```
```   468   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   469   assumes "convergent (\<lambda>n. X n)"
```
```   470   assumes "convergent (\<lambda>n. Y n)"
```
```   471   shows "convergent (\<lambda>n. X n + Y n)"
```
```   472   using assms unfolding convergent_def by (fast intro: tendsto_add)
```
```   473
```
```   474 lemma convergent_setsum:
```
```   475   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```   476   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```   477   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```   478 proof (cases "finite A")
```
```   479   case True from this and assms show ?thesis
```
```   480     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```   481 qed (simp add: convergent_const)
```
```   482
```
```   483 lemma (in bounded_linear) convergent:
```
```   484   assumes "convergent (\<lambda>n. X n)"
```
```   485   shows "convergent (\<lambda>n. f (X n))"
```
```   486   using assms unfolding convergent_def by (fast intro: tendsto)
```
```   487
```
```   488 lemma (in bounded_bilinear) convergent:
```
```   489   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```   490   shows "convergent (\<lambda>n. X n ** Y n)"
```
```   491   using assms unfolding convergent_def by (fast intro: tendsto)
```
```   492
```
```   493 lemma convergent_minus_iff:
```
```   494   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   495   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```   496 apply (simp add: convergent_def)
```
```   497 apply (auto dest: tendsto_minus)
```
```   498 apply (drule tendsto_minus, auto)
```
```   499 done
```
```   500
```
```   501 lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::real)) \<Longrightarrow> lim f \<le> x"
```
```   502   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
```
```   503
```
```   504 lemma monoseq_le:
```
```   505   "monoseq a \<Longrightarrow> a ----> (x::real) \<Longrightarrow>
```
```   506     ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```   507   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
```
```   508
```
```   509 lemma LIMSEQ_subseq_LIMSEQ:
```
```   510   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```   511   unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
```
```   512
```
```   513 lemma convergent_subseq_convergent:
```
```   514   "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```   515   unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
```
```   516
```
```   517
```
```   518 subsection {* Bounded Monotonic Sequences *}
```
```   519
```
```   520 text{*Bounded Sequence*}
```
```   521
```
```   522 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   523 by (simp add: Bseq_def)
```
```   524
```
```   525 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   526 by (auto simp add: Bseq_def)
```
```   527
```
```   528 lemma lemma_NBseq_def:
```
```   529   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   530 proof safe
```
```   531   fix K :: real
```
```   532   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```   533   then have "K \<le> real (Suc n)" by auto
```
```   534   moreover assume "\<forall>m. norm (X m) \<le> K"
```
```   535   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```   536     by (blast intro: order_trans)
```
```   537   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```   538 qed (force simp add: real_of_nat_Suc)
```
```   539
```
```   540 text{* alternative definition for Bseq *}
```
```   541 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   542 apply (simp add: Bseq_def)
```
```   543 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   544 done
```
```   545
```
```   546 lemma lemma_NBseq_def2:
```
```   547      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   548 apply (subst lemma_NBseq_def, auto)
```
```   549 apply (rule_tac x = "Suc N" in exI)
```
```   550 apply (rule_tac [2] x = N in exI)
```
```   551 apply (auto simp add: real_of_nat_Suc)
```
```   552  prefer 2 apply (blast intro: order_less_imp_le)
```
```   553 apply (drule_tac x = n in spec, simp)
```
```   554 done
```
```   555
```
```   556 (* yet another definition for Bseq *)
```
```   557 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   558 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   559
```
```   560 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```   561
```
```   562 text{*alternative formulation for boundedness*}
```
```   563 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```   564 apply (unfold Bseq_def, safe)
```
```   565 apply (rule_tac [2] x = "k + norm x" in exI)
```
```   566 apply (rule_tac x = K in exI, simp)
```
```   567 apply (rule exI [where x = 0], auto)
```
```   568 apply (erule order_less_le_trans, simp)
```
```   569 apply (drule_tac x=n in spec, fold diff_minus)
```
```   570 apply (drule order_trans [OF norm_triangle_ineq2])
```
```   571 apply simp
```
```   572 done
```
```   573
```
```   574 text{*alternative formulation for boundedness*}
```
```   575 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```   576 apply safe
```
```   577 apply (simp add: Bseq_def, safe)
```
```   578 apply (rule_tac x = "K + norm (X N)" in exI)
```
```   579 apply auto
```
```   580 apply (erule order_less_le_trans, simp)
```
```   581 apply (rule_tac x = N in exI, safe)
```
```   582 apply (drule_tac x = n in spec)
```
```   583 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```   584 apply (auto simp add: Bseq_iff2)
```
```   585 done
```
```   586
```
```   587 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```   588 apply (simp add: Bseq_def)
```
```   589 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```   590 apply (drule_tac x = n in spec, arith)
```
```   591 done
```
```   592
```
```   593 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   594
```
```   595 lemma Bseq_isUb:
```
```   596   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   597 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   598
```
```   599 text{* Use completeness of reals (supremum property)
```
```   600    to show that any bounded sequence has a least upper bound*}
```
```   601
```
```   602 lemma Bseq_isLub:
```
```   603   "!!(X::nat=>real). Bseq X ==>
```
```   604    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   605 by (blast intro: reals_complete Bseq_isUb)
```
```   606
```
```   607 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   608
```
```   609 (* TODO: delete *)
```
```   610 (* FIXME: one use in NSA/HSEQ.thy *)
```
```   611 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   612 unfolding tendsto_def eventually_sequentially
```
```   613 apply (rule_tac x = "X m" in exI, safe)
```
```   614 apply (rule_tac x = m in exI, safe)
```
```   615 apply (drule spec, erule impE, auto)
```
```   616 done
```
```   617
```
```   618 text {* A monotone sequence converges to its least upper bound. *}
```
```   619
```
```   620 lemma isLub_mono_imp_LIMSEQ:
```
```   621   fixes X :: "nat \<Rightarrow> real"
```
```   622   assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
```
```   623   assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
```
```   624   shows "X ----> u"
```
```   625 proof (rule LIMSEQ_I)
```
```   626   have 1: "\<forall>n. X n \<le> u"
```
```   627     using isLubD2 [OF u] by auto
```
```   628   have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
```
```   629     using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
```
```   630   hence 2: "\<forall>y<u. \<exists>n. y < X n"
```
```   631     by (metis not_le)
```
```   632   fix r :: real assume "0 < r"
```
```   633   hence "u - r < u" by simp
```
```   634   hence "\<exists>m. u - r < X m" using 2 by simp
```
```   635   then obtain m where "u - r < X m" ..
```
```   636   with X have "\<forall>n\<ge>m. u - r < X n"
```
```   637     by (fast intro: less_le_trans)
```
```   638   hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
```
```   639   thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
```
```   640     using 1 by (simp add: diff_less_eq add_commute)
```
```   641 qed
```
```   642
```
```   643 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```   644
```
```   645 lemma Bseq_mono_convergent:
```
```   646    "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
```
```   647   by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
```
```   648
```
```   649 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
```
```   650   by (simp add: Bseq_def)
```
```   651
```
```   652 text{*Main monotonicity theorem*}
```
```   653
```
```   654 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
```
```   655   by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
```
```   656             Bseq_mono_convergent)
```
```   657
```
```   658 subsubsection{*Increasing and Decreasing Series*}
```
```   659
```
```   660 lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::real)"
```
```   661   by (metis incseq_def LIMSEQ_le_const)
```
```   662
```
```   663 lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::real) \<le> X n"
```
```   664   by (metis decseq_def LIMSEQ_le_const2)
```
```   665
```
```   666 subsection {* Cauchy Sequences *}
```
```   667
```
```   668 lemma metric_CauchyI:
```
```   669   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```   670   by (simp add: Cauchy_def)
```
```   671
```
```   672 lemma metric_CauchyD:
```
```   673   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```   674   by (simp add: Cauchy_def)
```
```   675
```
```   676 lemma Cauchy_iff:
```
```   677   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   678   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```   679   unfolding Cauchy_def dist_norm ..
```
```   680
```
```   681 lemma Cauchy_iff2:
```
```   682   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```   683 apply (simp add: Cauchy_iff, auto)
```
```   684 apply (drule reals_Archimedean, safe)
```
```   685 apply (drule_tac x = n in spec, auto)
```
```   686 apply (rule_tac x = M in exI, auto)
```
```   687 apply (drule_tac x = m in spec, simp)
```
```   688 apply (drule_tac x = na in spec, auto)
```
```   689 done
```
```   690
```
```   691 lemma CauchyI:
```
```   692   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   693   shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
```
```   694 by (simp add: Cauchy_iff)
```
```   695
```
```   696 lemma CauchyD:
```
```   697   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   698   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```   699 by (simp add: Cauchy_iff)
```
```   700
```
```   701 lemma Cauchy_subseq_Cauchy:
```
```   702   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```   703 apply (auto simp add: Cauchy_def)
```
```   704 apply (drule_tac x=e in spec, clarify)
```
```   705 apply (rule_tac x=M in exI, clarify)
```
```   706 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```   707 done
```
```   708
```
```   709 subsubsection {* Cauchy Sequences are Bounded *}
```
```   710
```
```   711 text{*A Cauchy sequence is bounded -- this is the standard
```
```   712   proof mechanization rather than the nonstandard proof*}
```
```   713
```
```   714 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```   715           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```   716 apply (clarify, drule spec, drule (1) mp)
```
```   717 apply (simp only: norm_minus_commute)
```
```   718 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```   719 apply simp
```
```   720 done
```
```   721
```
```   722 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
```
```   723 apply (simp add: Cauchy_iff)
```
```   724 apply (drule spec, drule mp, rule zero_less_one, safe)
```
```   725 apply (drule_tac x="M" in spec, simp)
```
```   726 apply (drule lemmaCauchy)
```
```   727 apply (rule_tac k="M" in Bseq_offset)
```
```   728 apply (simp add: Bseq_def)
```
```   729 apply (rule_tac x="1 + norm (X M)" in exI)
```
```   730 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
```
```   731 apply (simp add: order_less_imp_le)
```
```   732 done
```
```   733
```
```   734 subsubsection {* Cauchy Sequences are Convergent *}
```
```   735
```
```   736 class complete_space = metric_space +
```
```   737   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```   738
```
```   739 class banach = real_normed_vector + complete_space
```
```   740
```
```   741 theorem LIMSEQ_imp_Cauchy:
```
```   742   assumes X: "X ----> a" shows "Cauchy X"
```
```   743 proof (rule metric_CauchyI)
```
```   744   fix e::real assume "0 < e"
```
```   745   hence "0 < e/2" by simp
```
```   746   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```   747   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```   748   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```   749   proof (intro exI allI impI)
```
```   750     fix m assume "N \<le> m"
```
```   751     hence m: "dist (X m) a < e/2" using N by fast
```
```   752     fix n assume "N \<le> n"
```
```   753     hence n: "dist (X n) a < e/2" using N by fast
```
```   754     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```   755       by (rule dist_triangle2)
```
```   756     also from m n have "\<dots> < e" by simp
```
```   757     finally show "dist (X m) (X n) < e" .
```
```   758   qed
```
```   759 qed
```
```   760
```
```   761 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```   762 unfolding convergent_def
```
```   763 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```   764
```
```   765 lemma Cauchy_convergent_iff:
```
```   766   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```   767   shows "Cauchy X = convergent X"
```
```   768 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```   769
```
```   770 text {*
```
```   771 Proof that Cauchy sequences converge based on the one from
```
```   772 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```   773 *}
```
```   774
```
```   775 text {*
```
```   776   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```   777   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```   778 *}
```
```   779
```
```   780 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```   781 by (simp add: isUbI setleI)
```
```   782
```
```   783 lemma real_Cauchy_convergent:
```
```   784   fixes X :: "nat \<Rightarrow> real"
```
```   785   assumes X: "Cauchy X"
```
```   786   shows "convergent X"
```
```   787 proof -
```
```   788   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```   789   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
```
```   790
```
```   791   { fix N x assume N: "\<forall>n\<ge>N. X n < x"
```
```   792   have "isUb UNIV S x"
```
```   793   proof (rule isUb_UNIV_I)
```
```   794   fix y::real assume "y \<in> S"
```
```   795   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```   796     by (simp add: S_def)
```
```   797   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```   798   hence "y < X (max M N)" by simp
```
```   799   also have "\<dots> < x" using N by simp
```
```   800   finally show "y \<le> x"
```
```   801     by (rule order_less_imp_le)
```
```   802   qed }
```
```   803   note bound_isUb = this
```
```   804
```
```   805   have "\<exists>u. isLub UNIV S u"
```
```   806   proof (rule reals_complete)
```
```   807   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
```
```   808     using CauchyD [OF X zero_less_one] by auto
```
```   809   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
```
```   810   show "\<exists>x. x \<in> S"
```
```   811   proof
```
```   812     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```   813       by (simp add: abs_diff_less_iff)
```
```   814     thus "X N - 1 \<in> S" by (rule mem_S)
```
```   815   qed
```
```   816   show "\<exists>u. isUb UNIV S u"
```
```   817   proof
```
```   818     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```   819       by (simp add: abs_diff_less_iff)
```
```   820     thus "isUb UNIV S (X N + 1)"
```
```   821       by (rule bound_isUb)
```
```   822   qed
```
```   823   qed
```
```   824   then obtain x where x: "isLub UNIV S x" ..
```
```   825   have "X ----> x"
```
```   826   proof (rule LIMSEQ_I)
```
```   827   fix r::real assume "0 < r"
```
```   828   hence r: "0 < r/2" by simp
```
```   829   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
```
```   830     using CauchyD [OF X r] by auto
```
```   831   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
```
```   832   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```   833     by (simp only: real_norm_def abs_diff_less_iff)
```
```   834
```
```   835   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```   836   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```   837   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```   838
```
```   839   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```   840   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```   841   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```   842
```
```   843   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
```
```   844   proof (intro exI allI impI)
```
```   845     fix n assume n: "N \<le> n"
```
```   846     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```   847     thus "norm (X n - x) < r" using 1 2
```
```   848       by (simp add: abs_diff_less_iff)
```
```   849   qed
```
```   850   qed
```
```   851   then show ?thesis unfolding convergent_def by auto
```
```   852 qed
```
```   853
```
```   854 instance real :: banach
```
```   855   by intro_classes (rule real_Cauchy_convergent)
```
```   856
```
```   857
```
```   858 subsection {* Power Sequences *}
```
```   859
```
```   860 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```   861 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```   862   also fact that bounded and monotonic sequence converges.*}
```
```   863
```
```   864 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```   865 apply (simp add: Bseq_def)
```
```   866 apply (rule_tac x = 1 in exI)
```
```   867 apply (simp add: power_abs)
```
```   868 apply (auto dest: power_mono)
```
```   869 done
```
```   870
```
```   871 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```   872 apply (clarify intro!: mono_SucI2)
```
```   873 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```   874 done
```
```   875
```
```   876 lemma convergent_realpow:
```
```   877   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```   878 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```   879
```
```   880 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```   881   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
```
```   882
```
```   883 lemma LIMSEQ_realpow_zero:
```
```   884   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```   885 proof cases
```
```   886   assume "0 \<le> x" and "x \<noteq> 0"
```
```   887   hence x0: "0 < x" by simp
```
```   888   assume x1: "x < 1"
```
```   889   from x0 x1 have "1 < inverse x"
```
```   890     by (rule one_less_inverse)
```
```   891   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```   892     by (rule LIMSEQ_inverse_realpow_zero)
```
```   893   thus ?thesis by (simp add: power_inverse)
```
```   894 qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
```
```   895
```
```   896 lemma LIMSEQ_power_zero:
```
```   897   fixes x :: "'a::{real_normed_algebra_1}"
```
```   898   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```   899 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```   900 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
```
```   901 apply (simp add: power_abs norm_power_ineq)
```
```   902 done
```
```   903
```
```   904 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
```
```   905   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
```
```   906
```
```   907 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```   908
```
```   909 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
```
```   910   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```   911
```
```   912 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
```
```   913   by (rule LIMSEQ_power_zero) simp
```
```   914
```
```   915 lemma tendsto_at_topI_sequentially:
```
```   916   fixes f :: "real \<Rightarrow> real"
```
```   917   assumes mono: "mono f"
```
```   918   assumes limseq: "(\<lambda>n. f (real n)) ----> y"
```
```   919   shows "(f ---> y) at_top"
```
```   920 proof (rule tendstoI)
```
```   921   fix e :: real assume "0 < e"
```
```   922   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```   923     by (auto simp: LIMSEQ_def dist_real_def)
```
```   924   { fix x :: real
```
```   925     from ex_le_of_nat[of x] guess n ..
```
```   926     note monoD[OF mono this]
```
```   927     also have "f (real_of_nat n) \<le> y"
```
```   928       by (rule LIMSEQ_le_const[OF limseq])
```
```   929          (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
```
```   930     finally have "f x \<le> y" . }
```
```   931   note le = this
```
```   932   have "eventually (\<lambda>x. real N \<le> x) at_top"
```
```   933     by (rule eventually_ge_at_top)
```
```   934   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
```
```   935   proof eventually_elim
```
```   936     fix x assume N': "real N \<le> x"
```
```   937     with N[of N] le have "y - f (real N) < e" by auto
```
```   938     moreover note monoD[OF mono N']
```
```   939     ultimately show "dist (f x) y < e"
```
```   940       using le[of x] by (auto simp: dist_real_def field_simps)
```
```   941   qed
```
```   942 qed
```
```   943
```
```   944 end
```