src/HOL/Lim.thy
author huffman
Tue Aug 16 09:31:23 2011 -0700 (2011-08-16)
changeset 44233 aa74ce315bae
parent 44218 f0e442e24816
child 44251 d101ed3177b6
permissions -rw-r--r--
add simp rules for isCont
paulson@10751
     1
(*  Title       : Lim.thy
paulson@10751
     2
    Author      : Jacques D. Fleuriot
paulson@10751
     3
    Copyright   : 1998  University of Cambridge
paulson@14477
     4
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
paulson@10751
     5
*)
paulson@10751
     6
huffman@21165
     7
header{* Limits and Continuity *}
paulson@10751
     8
nipkow@15131
     9
theory Lim
haftmann@29197
    10
imports SEQ
nipkow@15131
    11
begin
paulson@14477
    12
huffman@22641
    13
text{*Standard Definitions*}
paulson@10751
    14
huffman@36661
    15
abbreviation
huffman@36662
    16
  LIM :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a, 'b] \<Rightarrow> bool"
wenzelm@21404
    17
        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
huffman@36661
    18
  "f -- a --> L \<equiv> (f ---> L) (at a)"
paulson@10751
    19
wenzelm@21404
    20
definition
huffman@36662
    21
  isCont :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a] \<Rightarrow> bool" where
wenzelm@19765
    22
  "isCont f a = (f -- a --> (f a))"
paulson@10751
    23
wenzelm@21404
    24
definition
huffman@31338
    25
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
haftmann@37767
    26
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
paulson@10751
    27
huffman@31392
    28
subsection {* Limits of Functions *}
huffman@31349
    29
huffman@36661
    30
lemma LIM_def: "f -- a --> L =
huffman@36661
    31
     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
huffman@36661
    32
        --> dist (f x) L < r)"
huffman@36661
    33
unfolding tendsto_iff eventually_at ..
paulson@10751
    34
huffman@31338
    35
lemma metric_LIM_I:
huffman@31338
    36
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
huffman@31338
    37
    \<Longrightarrow> f -- a --> L"
huffman@31338
    38
by (simp add: LIM_def)
huffman@31338
    39
huffman@31338
    40
lemma metric_LIM_D:
huffman@31338
    41
  "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
huffman@31338
    42
    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
huffman@31338
    43
by (simp add: LIM_def)
paulson@14477
    44
paulson@14477
    45
lemma LIM_eq:
huffman@31338
    46
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
huffman@31338
    47
  shows "f -- a --> L =
huffman@20561
    48
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
huffman@31338
    49
by (simp add: LIM_def dist_norm)
paulson@14477
    50
huffman@20552
    51
lemma LIM_I:
huffman@31338
    52
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
huffman@31338
    53
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
huffman@20552
    54
      ==> f -- a --> L"
huffman@20552
    55
by (simp add: LIM_eq)
huffman@20552
    56
paulson@14477
    57
lemma LIM_D:
huffman@31338
    58
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
huffman@31338
    59
  shows "[| f -- a --> L; 0<r |]
huffman@20561
    60
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
paulson@14477
    61
by (simp add: LIM_eq)
paulson@14477
    62
huffman@31338
    63
lemma LIM_offset:
huffman@36662
    64
  fixes a :: "'a::real_normed_vector"
huffman@31338
    65
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
huffman@36662
    66
apply (rule topological_tendstoI)
huffman@36662
    67
apply (drule (2) topological_tendstoD)
huffman@36662
    68
apply (simp only: eventually_at dist_norm)
huffman@36662
    69
apply (clarify, rule_tac x=d in exI, safe)
huffman@20756
    70
apply (drule_tac x="x + k" in spec)
nipkow@29667
    71
apply (simp add: algebra_simps)
huffman@20756
    72
done
huffman@20756
    73
huffman@31338
    74
lemma LIM_offset_zero:
huffman@36662
    75
  fixes a :: "'a::real_normed_vector"
huffman@31338
    76
  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
huffman@21239
    77
by (drule_tac k="a" in LIM_offset, simp add: add_commute)
huffman@21239
    78
huffman@31338
    79
lemma LIM_offset_zero_cancel:
huffman@36662
    80
  fixes a :: "'a::real_normed_vector"
huffman@31338
    81
  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
huffman@21239
    82
by (drule_tac k="- a" in LIM_offset, simp)
huffman@21239
    83
paulson@15228
    84
lemma LIM_const [simp]: "(%x. k) -- x --> k"
huffman@36661
    85
by (rule tendsto_const)
paulson@14477
    86
hoelzl@32650
    87
lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
hoelzl@32650
    88
paulson@14477
    89
lemma LIM_add:
huffman@36662
    90
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
paulson@14477
    91
  assumes f: "f -- a --> L" and g: "g -- a --> M"
huffman@31338
    92
  shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
huffman@36661
    93
using assms by (rule tendsto_add)
paulson@14477
    94
huffman@21257
    95
lemma LIM_add_zero:
huffman@36662
    96
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
    97
  shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
huffman@44194
    98
  by (rule tendsto_add_zero)
huffman@21257
    99
huffman@31338
   100
lemma LIM_minus:
huffman@36662
   101
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   102
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
huffman@36661
   103
by (rule tendsto_minus)
paulson@14477
   104
huffman@31338
   105
(* TODO: delete *)
paulson@14477
   106
lemma LIM_add_minus:
huffman@36662
   107
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   108
  shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
huffman@20552
   109
by (intro LIM_add LIM_minus)
paulson@14477
   110
paulson@14477
   111
lemma LIM_diff:
huffman@36662
   112
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   113
  shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
huffman@36661
   114
by (rule tendsto_diff)
paulson@14477
   115
huffman@31338
   116
lemma LIM_zero:
huffman@36662
   117
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   118
  shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
huffman@36662
   119
unfolding tendsto_iff dist_norm by simp
huffman@21239
   120
huffman@31338
   121
lemma LIM_zero_cancel:
huffman@36662
   122
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   123
  shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
huffman@36662
   124
unfolding tendsto_iff dist_norm by simp
huffman@21239
   125
huffman@31338
   126
lemma LIM_zero_iff:
huffman@31338
   127
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   128
  shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
huffman@36662
   129
unfolding tendsto_iff dist_norm by simp
huffman@21399
   130
huffman@31338
   131
lemma metric_LIM_imp_LIM:
huffman@21257
   132
  assumes f: "f -- a --> l"
huffman@31338
   133
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
huffman@21257
   134
  shows "g -- a --> m"
huffman@36662
   135
apply (rule tendstoI, drule tendstoD [OF f])
huffman@36662
   136
apply (simp add: eventually_at_topological, safe)
huffman@36662
   137
apply (rule_tac x="S" in exI, safe)
huffman@36662
   138
apply (drule_tac x="x" in bspec, safe)
huffman@21257
   139
apply (erule (1) order_le_less_trans [OF le])
huffman@21257
   140
done
huffman@21257
   141
huffman@31338
   142
lemma LIM_imp_LIM:
huffman@36662
   143
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@36662
   144
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
huffman@31338
   145
  assumes f: "f -- a --> l"
huffman@31338
   146
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
huffman@31338
   147
  shows "g -- a --> m"
huffman@31338
   148
apply (rule metric_LIM_imp_LIM [OF f])
huffman@31338
   149
apply (simp add: dist_norm le)
huffman@31338
   150
done
huffman@31338
   151
huffman@31338
   152
lemma LIM_norm:
huffman@36662
   153
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   154
  shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
huffman@36661
   155
by (rule tendsto_norm)
huffman@21257
   156
huffman@31338
   157
lemma LIM_norm_zero:
huffman@36662
   158
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   159
  shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
huffman@36662
   160
by (rule tendsto_norm_zero)
huffman@21257
   161
huffman@31338
   162
lemma LIM_norm_zero_cancel:
huffman@36662
   163
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   164
  shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
huffman@36662
   165
by (rule tendsto_norm_zero_cancel)
huffman@21257
   166
huffman@31338
   167
lemma LIM_norm_zero_iff:
huffman@36662
   168
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   169
  shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
huffman@36662
   170
by (rule tendsto_norm_zero_iff)
huffman@21399
   171
huffman@22627
   172
lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
huffman@44194
   173
  by (rule tendsto_rabs)
huffman@22627
   174
huffman@22627
   175
lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
huffman@44194
   176
  by (rule tendsto_rabs_zero)
huffman@22627
   177
huffman@22627
   178
lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
huffman@44194
   179
  by (rule tendsto_rabs_zero_cancel)
huffman@22627
   180
huffman@22627
   181
lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
huffman@44194
   182
  by (rule tendsto_rabs_zero_iff)
huffman@22627
   183
huffman@44205
   184
lemma trivial_limit_at:
huffman@36662
   185
  fixes a :: "'a::real_normed_algebra_1"
huffman@44205
   186
  shows "\<not> trivial_limit (at a)"  -- {* TODO: find a more appropriate class *}
huffman@44205
   187
unfolding trivial_limit_def
huffman@36662
   188
unfolding eventually_at dist_norm
huffman@36662
   189
by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
huffman@36662
   190
huffman@20561
   191
lemma LIM_const_not_eq:
huffman@23076
   192
  fixes a :: "'a::real_normed_algebra_1"
huffman@44205
   193
  fixes k L :: "'b::t2_space"
huffman@23076
   194
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
huffman@44205
   195
by (simp add: tendsto_const_iff trivial_limit_at)
paulson@14477
   196
huffman@21786
   197
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
huffman@21786
   198
huffman@20561
   199
lemma LIM_const_eq:
huffman@23076
   200
  fixes a :: "'a::real_normed_algebra_1"
huffman@44205
   201
  fixes k L :: "'b::t2_space"
huffman@23076
   202
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
huffman@44205
   203
  by (simp add: tendsto_const_iff trivial_limit_at)
paulson@14477
   204
huffman@20561
   205
lemma LIM_unique:
huffman@31338
   206
  fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
huffman@44205
   207
  fixes L M :: "'b::t2_space"
huffman@23076
   208
  shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
huffman@44205
   209
  using trivial_limit_at by (rule tendsto_unique)
paulson@14477
   210
huffman@23069
   211
lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
huffman@36661
   212
by (rule tendsto_ident_at)
paulson@14477
   213
paulson@14477
   214
text{*Limits are equal for functions equal except at limit point*}
paulson@14477
   215
lemma LIM_equal:
paulson@14477
   216
     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
huffman@36662
   217
unfolding tendsto_def eventually_at_topological by simp
paulson@14477
   218
huffman@20796
   219
lemma LIM_cong:
huffman@20796
   220
  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
huffman@21399
   221
   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
huffman@36662
   222
by (simp add: LIM_equal)
huffman@20796
   223
huffman@31338
   224
lemma metric_LIM_equal2:
huffman@21282
   225
  assumes 1: "0 < R"
huffman@31338
   226
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
huffman@21282
   227
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
huffman@36662
   228
apply (rule topological_tendstoI)
huffman@36662
   229
apply (drule (2) topological_tendstoD)
huffman@36662
   230
apply (simp add: eventually_at, safe)
huffman@36662
   231
apply (rule_tac x="min d R" in exI, safe)
huffman@21282
   232
apply (simp add: 1)
huffman@21282
   233
apply (simp add: 2)
huffman@21282
   234
done
huffman@21282
   235
huffman@31338
   236
lemma LIM_equal2:
huffman@36662
   237
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   238
  assumes 1: "0 < R"
huffman@31338
   239
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
huffman@31338
   240
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
huffman@36662
   241
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
huffman@31338
   242
huffman@36662
   243
text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
paulson@14477
   244
lemma LIM_trans:
huffman@31338
   245
  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   246
  shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
paulson@14477
   247
apply (drule LIM_add, assumption)
paulson@14477
   248
apply (auto simp add: add_assoc)
paulson@14477
   249
done
paulson@14477
   250
huffman@21239
   251
lemma LIM_compose:
huffman@21239
   252
  assumes g: "g -- l --> g l"
huffman@21239
   253
  assumes f: "f -- a --> l"
huffman@21239
   254
  shows "(\<lambda>x. g (f x)) -- a --> g l"
huffman@44218
   255
  using assms by (rule tendsto_compose)
huffman@21239
   256
huffman@36662
   257
lemma LIM_compose_eventually:
huffman@36662
   258
  assumes f: "f -- a --> b"
huffman@36662
   259
  assumes g: "g -- b --> c"
huffman@36662
   260
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
huffman@36662
   261
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@36662
   262
proof (rule topological_tendstoI)
huffman@36662
   263
  fix C assume C: "open C" "c \<in> C"
huffman@36662
   264
  obtain B where B: "open B" "b \<in> B"
huffman@36662
   265
    and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> b \<Longrightarrow> g y \<in> C"
huffman@36662
   266
    using topological_tendstoD [OF g C]
huffman@36662
   267
    unfolding eventually_at_topological by fast
huffman@36662
   268
  obtain A where A: "open A" "a \<in> A"
huffman@36662
   269
    and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
huffman@36662
   270
    using topological_tendstoD [OF f B]
huffman@36662
   271
    unfolding eventually_at_topological by fast
huffman@36662
   272
  have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> g (f x) \<in> C) (at a)"
huffman@36662
   273
  unfolding eventually_at_topological
huffman@36662
   274
  proof (intro exI conjI ballI impI)
huffman@36662
   275
    show "open A" and "a \<in> A" using A .
huffman@21239
   276
  next
huffman@36662
   277
    fix x assume "x \<in> A" and "x \<noteq> a" and "f x \<noteq> b"
huffman@36662
   278
    then show "g (f x) \<in> C" by (simp add: gC fB)
huffman@21239
   279
  qed
huffman@36662
   280
  with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
huffman@36662
   281
    by (rule eventually_rev_mp)
huffman@21239
   282
qed
huffman@21239
   283
huffman@31338
   284
lemma metric_LIM_compose2:
huffman@31338
   285
  assumes f: "f -- a --> b"
huffman@31338
   286
  assumes g: "g -- b --> c"
huffman@31338
   287
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
huffman@31338
   288
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@36662
   289
using f g inj [folded eventually_at]
huffman@36662
   290
by (rule LIM_compose_eventually)
huffman@31338
   291
huffman@23040
   292
lemma LIM_compose2:
huffman@31338
   293
  fixes a :: "'a::real_normed_vector"
huffman@23040
   294
  assumes f: "f -- a --> b"
huffman@23040
   295
  assumes g: "g -- b --> c"
huffman@23040
   296
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
huffman@23040
   297
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@31338
   298
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
huffman@23040
   299
huffman@21239
   300
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
huffman@21239
   301
unfolding o_def by (rule LIM_compose)
huffman@21239
   302
huffman@21282
   303
lemma real_LIM_sandwich_zero:
huffman@36662
   304
  fixes f g :: "'a::topological_space \<Rightarrow> real"
huffman@21282
   305
  assumes f: "f -- a --> 0"
huffman@21282
   306
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
huffman@21282
   307
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
huffman@21282
   308
  shows "g -- a --> 0"
huffman@21282
   309
proof (rule LIM_imp_LIM [OF f])
huffman@21282
   310
  fix x assume x: "x \<noteq> a"
huffman@21282
   311
  have "norm (g x - 0) = g x" by (simp add: 1 x)
huffman@21282
   312
  also have "g x \<le> f x" by (rule 2 [OF x])
huffman@21282
   313
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
huffman@21282
   314
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
huffman@21282
   315
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
huffman@21282
   316
qed
huffman@21282
   317
huffman@22442
   318
text {* Bounded Linear Operators *}
huffman@21282
   319
huffman@21282
   320
lemma (in bounded_linear) LIM:
huffman@21282
   321
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@36661
   322
by (rule tendsto)
huffman@31349
   323
huffman@21282
   324
lemma (in bounded_linear) LIM_zero:
huffman@21282
   325
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@44194
   326
  by (rule tendsto_zero)
huffman@21282
   327
huffman@22442
   328
text {* Bounded Bilinear Operators *}
huffman@21282
   329
huffman@31349
   330
lemma (in bounded_bilinear) LIM:
huffman@31349
   331
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@36661
   332
by (rule tendsto)
huffman@31349
   333
huffman@21282
   334
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@31338
   335
  fixes a :: "'d::metric_space"
huffman@21282
   336
  assumes f: "f -- a --> 0"
huffman@21282
   337
  assumes g: "g -- a --> 0"
huffman@21282
   338
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@44194
   339
  using f g by (rule tendsto_zero)
huffman@21282
   340
huffman@21282
   341
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   342
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@44194
   343
  by (rule tendsto_left_zero)
huffman@21282
   344
huffman@21282
   345
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   346
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@44194
   347
  by (rule tendsto_right_zero)
huffman@21282
   348
huffman@23127
   349
lemmas LIM_mult = mult.LIM
huffman@21282
   350
huffman@23127
   351
lemmas LIM_mult_zero = mult.LIM_prod_zero
huffman@21282
   352
huffman@23127
   353
lemmas LIM_mult_left_zero = mult.LIM_left_zero
huffman@21282
   354
huffman@23127
   355
lemmas LIM_mult_right_zero = mult.LIM_right_zero
huffman@21282
   356
huffman@23127
   357
lemmas LIM_scaleR = scaleR.LIM
huffman@21282
   358
huffman@23127
   359
lemmas LIM_of_real = of_real.LIM
huffman@22627
   360
huffman@22627
   361
lemma LIM_power:
huffman@36665
   362
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   363
  assumes f: "f -- a --> l"
huffman@22627
   364
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
huffman@44194
   365
  using assms by (rule tendsto_power)
huffman@22627
   366
huffman@31355
   367
lemma LIM_inverse:
huffman@31355
   368
  fixes L :: "'a::real_normed_div_algebra"
huffman@31355
   369
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
huffman@31355
   370
by (rule tendsto_inverse)
huffman@22637
   371
huffman@22637
   372
lemma LIM_inverse_fun:
huffman@22637
   373
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
huffman@22637
   374
  shows "inverse -- a --> inverse a"
huffman@31355
   375
by (rule LIM_inverse [OF LIM_ident a])
huffman@22637
   376
huffman@29885
   377
lemma LIM_sgn:
huffman@36665
   378
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   379
  shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
huffman@44194
   380
  by (rule tendsto_sgn)
huffman@29885
   381
paulson@14477
   382
huffman@20755
   383
subsection {* Continuity *}
paulson@14477
   384
huffman@31338
   385
lemma LIM_isCont_iff:
huffman@36665
   386
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   387
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   388
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   389
huffman@31338
   390
lemma isCont_iff:
huffman@36665
   391
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   392
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   393
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   394
huffman@23069
   395
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
huffman@23069
   396
  unfolding isCont_def by (rule LIM_ident)
huffman@21239
   397
huffman@21786
   398
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   399
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   400
huffman@44233
   401
lemma isCont_norm [simp]:
huffman@36665
   402
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   403
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   404
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   405
huffman@44233
   406
lemma isCont_rabs [simp]:
huffman@44233
   407
  fixes f :: "'a::topological_space \<Rightarrow> real"
huffman@44233
   408
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
huffman@22627
   409
  unfolding isCont_def by (rule LIM_rabs)
huffman@22627
   410
huffman@44233
   411
lemma isCont_add [simp]:
huffman@36665
   412
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   413
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   414
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   415
huffman@44233
   416
lemma isCont_minus [simp]:
huffman@36665
   417
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   418
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   419
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   420
huffman@44233
   421
lemma isCont_diff [simp]:
huffman@36665
   422
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   423
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   424
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   425
huffman@44233
   426
lemma isCont_mult [simp]:
huffman@36665
   427
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   428
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   429
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   430
huffman@44233
   431
lemma isCont_inverse [simp]:
huffman@36665
   432
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   433
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   434
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   435
huffman@44233
   436
lemma isCont_divide [simp]:
huffman@44233
   437
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
huffman@44233
   438
  shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
huffman@44233
   439
  unfolding isCont_def by (rule tendsto_divide)
huffman@44233
   440
huffman@21239
   441
lemma isCont_LIM_compose:
huffman@21239
   442
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   443
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   444
huffman@31338
   445
lemma metric_isCont_LIM_compose2:
huffman@31338
   446
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@31338
   447
  assumes g: "g -- f a --> l"
huffman@31338
   448
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
huffman@31338
   449
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@31338
   450
by (rule metric_LIM_compose2 [OF f g inj])
huffman@31338
   451
huffman@23040
   452
lemma isCont_LIM_compose2:
huffman@31338
   453
  fixes a :: "'a::real_normed_vector"
huffman@23040
   454
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@23040
   455
  assumes g: "g -- f a --> l"
huffman@23040
   456
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
huffman@23040
   457
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@23040
   458
by (rule LIM_compose2 [OF f g inj])
huffman@23040
   459
huffman@21239
   460
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   461
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   462
huffman@21239
   463
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   464
  unfolding o_def by (rule isCont_o2)
huffman@21282
   465
huffman@44233
   466
lemma (in bounded_linear) isCont:
huffman@44233
   467
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
huffman@44233
   468
  unfolding isCont_def by (rule LIM)
huffman@21282
   469
huffman@21282
   470
lemma (in bounded_bilinear) isCont:
huffman@21282
   471
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   472
  unfolding isCont_def by (rule LIM)
huffman@21282
   473
huffman@44233
   474
lemmas isCont_scaleR [simp] = scaleR.isCont
huffman@21239
   475
huffman@44233
   476
lemma isCont_of_real [simp]:
huffman@31338
   477
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
huffman@44233
   478
  by (rule of_real.isCont)
huffman@22627
   479
huffman@44233
   480
lemma isCont_power [simp]:
huffman@36665
   481
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   482
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
huffman@22627
   483
  unfolding isCont_def by (rule LIM_power)
huffman@22627
   484
huffman@44233
   485
lemma isCont_sgn [simp]:
huffman@36665
   486
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   487
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
huffman@29885
   488
  unfolding isCont_def by (rule LIM_sgn)
huffman@29885
   489
huffman@44233
   490
lemma isCont_setsum [simp]:
huffman@44233
   491
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
huffman@44233
   492
  fixes A :: "'a set"
huffman@44233
   493
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
huffman@44233
   494
  unfolding isCont_def by (simp add: tendsto_setsum)
paulson@15228
   495
huffman@44233
   496
lemmas isCont_intros =
huffman@44233
   497
  isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
huffman@44233
   498
  isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
huffman@44233
   499
  isCont_of_real isCont_power isCont_sgn isCont_setsum
hoelzl@29803
   500
hoelzl@29803
   501
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
hoelzl@29803
   502
  and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
hoelzl@29803
   503
  shows "0 \<le> f x"
hoelzl@29803
   504
proof (rule ccontr)
hoelzl@29803
   505
  assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
hoelzl@29803
   506
  hence "0 < - f x / 2" by auto
hoelzl@29803
   507
  from isCont[unfolded isCont_def, THEN LIM_D, OF this]
hoelzl@29803
   508
  obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x' - x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x' - f x \<bar> < - f x / 2" by auto
hoelzl@29803
   509
hoelzl@29803
   510
  let ?x = "x - min (s / 2) ((x - b) / 2)"
hoelzl@29803
   511
  have "?x < x" and "\<bar> ?x - x \<bar> < s"
hoelzl@29803
   512
    using `b < x` and `0 < s` by auto
hoelzl@29803
   513
  have "b < ?x"
hoelzl@29803
   514
  proof (cases "s < x - b")
hoelzl@29803
   515
    case True thus ?thesis using `0 < s` by auto
hoelzl@29803
   516
  next
hoelzl@29803
   517
    case False hence "s / 2 \<ge> (x - b) / 2" by auto
haftmann@32642
   518
    hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
hoelzl@29803
   519
    thus ?thesis using `b < x` by auto
hoelzl@29803
   520
  qed
hoelzl@29803
   521
  hence "0 \<le> f ?x" using all_le `?x < x` by auto
hoelzl@29803
   522
  moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
hoelzl@29803
   523
    using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
hoelzl@29803
   524
  hence "f ?x - f x < - f x / 2" by auto
hoelzl@29803
   525
  hence "f ?x < f x / 2" by auto
hoelzl@29803
   526
  hence "f ?x < 0" using `f x < 0` by auto
hoelzl@29803
   527
  thus False using `0 \<le> f ?x` by auto
hoelzl@29803
   528
qed
huffman@31338
   529
paulson@14477
   530
huffman@20755
   531
subsection {* Uniform Continuity *}
huffman@20755
   532
paulson@14477
   533
lemma isUCont_isCont: "isUCont f ==> isCont f x"
huffman@23012
   534
by (simp add: isUCont_def isCont_def LIM_def, force)
paulson@14477
   535
huffman@23118
   536
lemma isUCont_Cauchy:
huffman@23118
   537
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   538
unfolding isUCont_def
huffman@31338
   539
apply (rule metric_CauchyI)
huffman@23118
   540
apply (drule_tac x=e in spec, safe)
huffman@31338
   541
apply (drule_tac e=s in metric_CauchyD, safe)
huffman@23118
   542
apply (rule_tac x=M in exI, simp)
huffman@23118
   543
done
huffman@23118
   544
huffman@23118
   545
lemma (in bounded_linear) isUCont: "isUCont f"
huffman@31338
   546
unfolding isUCont_def dist_norm
huffman@23118
   547
proof (intro allI impI)
huffman@23118
   548
  fix r::real assume r: "0 < r"
huffman@23118
   549
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
huffman@23118
   550
    using pos_bounded by fast
huffman@23118
   551
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@23118
   552
  proof (rule exI, safe)
huffman@23118
   553
    from r K show "0 < r / K" by (rule divide_pos_pos)
huffman@23118
   554
  next
huffman@23118
   555
    fix x y :: 'a
huffman@23118
   556
    assume xy: "norm (x - y) < r / K"
huffman@23118
   557
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
huffman@23118
   558
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
huffman@23118
   559
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@23118
   560
    finally show "norm (f x - f y) < r" .
huffman@23118
   561
  qed
huffman@23118
   562
qed
huffman@23118
   563
huffman@23118
   564
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   565
by (rule isUCont [THEN isUCont_Cauchy])
huffman@23118
   566
paulson@14477
   567
huffman@21165
   568
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   569
kleing@19023
   570
lemma LIMSEQ_SEQ_conv1:
huffman@36662
   571
  fixes a :: "'a::metric_space" and L :: "'b::metric_space"
huffman@21165
   572
  assumes X: "X -- a --> L"
kleing@19023
   573
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@31338
   574
proof (safe intro!: metric_LIMSEQ_I)
huffman@21165
   575
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   576
  fix r :: real
huffman@21165
   577
  assume rgz: "0 < r"
huffman@21165
   578
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   579
  assume S: "S ----> a"
huffman@31338
   580
  from metric_LIM_D [OF X rgz] obtain s
huffman@21165
   581
    where sgz: "0 < s"
huffman@31338
   582
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
huffman@21165
   583
    by fast
huffman@31338
   584
  from metric_LIMSEQ_D [OF S sgz]
huffman@31338
   585
  obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
huffman@31338
   586
  hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
huffman@31338
   587
  thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
kleing@19023
   588
qed
kleing@19023
   589
huffman@31338
   590
kleing@19023
   591
lemma LIMSEQ_SEQ_conv2:
huffman@36662
   592
  fixes a :: real and L :: "'a::metric_space"
kleing@19023
   593
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   594
  shows "X -- a --> L"
kleing@19023
   595
proof (rule ccontr)
kleing@19023
   596
  assume "\<not> (X -- a --> L)"
huffman@31338
   597
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
huffman@31338
   598
    unfolding LIM_def dist_norm .
huffman@31338
   599
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> dist (X x) L < r)" by simp
huffman@31338
   600
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less)
huffman@31338
   601
  then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto
kleing@19023
   602
huffman@31338
   603
  let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
huffman@31338
   604
  have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
huffman@21165
   605
    using rdef by simp
huffman@31338
   606
  hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
huffman@21165
   607
    by (rule someI_ex)
huffman@21165
   608
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   609
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@31338
   610
    and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
huffman@21165
   611
    by fast+
huffman@21165
   612
kleing@19023
   613
  have "?F ----> a"
huffman@21165
   614
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   615
      fix e::real
kleing@19023
   616
      assume "0 < e"
kleing@19023
   617
        (* choose no such that inverse (real (Suc n)) < e *)
huffman@23441
   618
      then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   619
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   620
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   621
      proof (intro exI allI impI)
kleing@19023
   622
        fix n
kleing@19023
   623
        assume mlen: "m \<le> n"
huffman@21165
   624
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   625
          by (rule F2)
huffman@21165
   626
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
huffman@23441
   627
          using mlen by auto
huffman@21165
   628
        also from nodef have
kleing@19023
   629
          "inverse (real (Suc m)) < e" .
huffman@21165
   630
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   631
      qed
kleing@19023
   632
  qed
kleing@19023
   633
  
kleing@19023
   634
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   635
    by (rule allI) (rule F1)
huffman@21165
   636
wenzelm@41550
   637
  moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
kleing@19023
   638
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   639
  
kleing@19023
   640
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   641
  proof -
kleing@19023
   642
    {
kleing@19023
   643
      fix no::nat
kleing@19023
   644
      obtain n where "n = no + 1" by simp
kleing@19023
   645
      then have nolen: "no \<le> n" by simp
kleing@19023
   646
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
huffman@31338
   647
      have "dist (X (?F n)) L \<ge> r"
huffman@21165
   648
        by (rule F3)
huffman@31338
   649
      with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
kleing@19023
   650
    }
huffman@31338
   651
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
huffman@31338
   652
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
huffman@31338
   653
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
kleing@19023
   654
  qed
kleing@19023
   655
  ultimately show False by simp
kleing@19023
   656
qed
kleing@19023
   657
kleing@19023
   658
lemma LIMSEQ_SEQ_conv:
huffman@20561
   659
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@36662
   660
   (X -- a --> (L::'a::metric_space))"
kleing@19023
   661
proof
kleing@19023
   662
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@23441
   663
  thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   664
next
kleing@19023
   665
  assume "(X -- a --> L)"
huffman@23441
   666
  thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
kleing@19023
   667
qed
kleing@19023
   668
paulson@10751
   669
end