src/HOL/Lim.thy
 author huffman Sun May 31 21:59:33 2009 -0700 (2009-05-31) changeset 31349 2261c8781f73 parent 31338 d41a8ba25b67 child 31353 14a58e2ca374 permissions -rw-r--r--
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
```     1 (*  Title       : Lim.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     5 *)
```
```     6
```
```     7 header{* Limits and Continuity *}
```
```     8
```
```     9 theory Lim
```
```    10 imports SEQ
```
```    11 begin
```
```    12
```
```    13 text{*Standard Definitions*}
```
```    14
```
```    15 definition
```
```    16   at :: "'a::metric_space \<Rightarrow> 'a filter" where
```
```    17   "at a = Abs_filter (\<lambda>P. \<exists>r>0. \<forall>x. x \<noteq> a \<and> dist x a < r \<longrightarrow> P x)"
```
```    18
```
```    19 definition
```
```    20   LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"
```
```    21         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
```
```    22   [code del]: "f -- a --> L =
```
```    23      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```    24         --> dist (f x) L < r)"
```
```    25
```
```    26 definition
```
```    27   isCont :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a] \<Rightarrow> bool" where
```
```    28   "isCont f a = (f -- a --> (f a))"
```
```    29
```
```    30 definition
```
```    31   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```    32   [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```    33
```
```    34 subsection {* Neighborhood Filter *}
```
```    35
```
```    36 lemma eventually_at:
```
```    37   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```    38 unfolding at_def
```
```    39 apply (rule eventually_Abs_filter)
```
```    40 apply (rule_tac x=1 in exI, simp)
```
```    41 apply (clarify, rule_tac x=r in exI, simp)
```
```    42 apply (clarify, rename_tac r s)
```
```    43 apply (rule_tac x="min r s" in exI, simp)
```
```    44 done
```
```    45
```
```    46 lemma LIM_conv_tendsto: "(f -- a --> L) \<longleftrightarrow> tendsto f L (at a)"
```
```    47 unfolding LIM_def tendsto_def eventually_at ..
```
```    48
```
```    49 subsection {* Limits of Functions *}
```
```    50
```
```    51 lemma metric_LIM_I:
```
```    52   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```    53     \<Longrightarrow> f -- a --> L"
```
```    54 by (simp add: LIM_def)
```
```    55
```
```    56 lemma metric_LIM_D:
```
```    57   "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
```
```    58     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```    59 by (simp add: LIM_def)
```
```    60
```
```    61 lemma LIM_eq:
```
```    62   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    63   shows "f -- a --> L =
```
```    64      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```    65 by (simp add: LIM_def dist_norm)
```
```    66
```
```    67 lemma LIM_I:
```
```    68   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    69   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```    70       ==> f -- a --> L"
```
```    71 by (simp add: LIM_eq)
```
```    72
```
```    73 lemma LIM_D:
```
```    74   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    75   shows "[| f -- a --> L; 0<r |]
```
```    76       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```    77 by (simp add: LIM_eq)
```
```    78
```
```    79 lemma LIM_offset:
```
```    80   fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
```
```    81   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```    82 unfolding LIM_def dist_norm
```
```    83 apply clarify
```
```    84 apply (drule_tac x="r" in spec, safe)
```
```    85 apply (rule_tac x="s" in exI, safe)
```
```    86 apply (drule_tac x="x + k" in spec)
```
```    87 apply (simp add: algebra_simps)
```
```    88 done
```
```    89
```
```    90 lemma LIM_offset_zero:
```
```    91   fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
```
```    92   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```    93 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
```
```    94
```
```    95 lemma LIM_offset_zero_cancel:
```
```    96   fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
```
```    97   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```    98 by (drule_tac k="- a" in LIM_offset, simp)
```
```    99
```
```   100 lemma LIM_const [simp]: "(%x. k) -- x --> k"
```
```   101 by (simp add: LIM_def)
```
```   102
```
```   103 lemma LIM_add:
```
```   104   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   105   assumes f: "f -- a --> L" and g: "g -- a --> M"
```
```   106   shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
```
```   107 using assms unfolding LIM_conv_tendsto by (rule tendsto_add)
```
```   108
```
```   109 lemma LIM_add_zero:
```
```   110   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   111   shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
```
```   112 by (drule (1) LIM_add, simp)
```
```   113
```
```   114 lemma minus_diff_minus:
```
```   115   fixes a b :: "'a::ab_group_add"
```
```   116   shows "(- a) - (- b) = - (a - b)"
```
```   117 by simp
```
```   118
```
```   119 lemma LIM_minus:
```
```   120   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   121   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
```
```   122 unfolding LIM_conv_tendsto by (rule tendsto_minus)
```
```   123
```
```   124 (* TODO: delete *)
```
```   125 lemma LIM_add_minus:
```
```   126   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   127   shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
```
```   128 by (intro LIM_add LIM_minus)
```
```   129
```
```   130 lemma LIM_diff:
```
```   131   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   132   shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
```
```   133 unfolding LIM_conv_tendsto by (rule tendsto_diff)
```
```   134
```
```   135 lemma LIM_zero:
```
```   136   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   137   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
```
```   138 by (simp add: LIM_def dist_norm)
```
```   139
```
```   140 lemma LIM_zero_cancel:
```
```   141   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   142   shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
```
```   143 by (simp add: LIM_def dist_norm)
```
```   144
```
```   145 lemma LIM_zero_iff:
```
```   146   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   147   shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
```
```   148 by (simp add: LIM_def dist_norm)
```
```   149
```
```   150 lemma metric_LIM_imp_LIM:
```
```   151   assumes f: "f -- a --> l"
```
```   152   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```   153   shows "g -- a --> m"
```
```   154 apply (rule metric_LIM_I, drule metric_LIM_D [OF f], safe)
```
```   155 apply (rule_tac x="s" in exI, safe)
```
```   156 apply (drule_tac x="x" in spec, safe)
```
```   157 apply (erule (1) order_le_less_trans [OF le])
```
```   158 done
```
```   159
```
```   160 lemma LIM_imp_LIM:
```
```   161   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   162   fixes g :: "'a::metric_space \<Rightarrow> 'c::real_normed_vector"
```
```   163   assumes f: "f -- a --> l"
```
```   164   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```   165   shows "g -- a --> m"
```
```   166 apply (rule metric_LIM_imp_LIM [OF f])
```
```   167 apply (simp add: dist_norm le)
```
```   168 done
```
```   169
```
```   170 lemma LIM_norm:
```
```   171   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   172   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
```
```   173 unfolding LIM_conv_tendsto by (rule tendsto_norm)
```
```   174
```
```   175 lemma LIM_norm_zero:
```
```   176   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   177   shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
```
```   178 by (drule LIM_norm, simp)
```
```   179
```
```   180 lemma LIM_norm_zero_cancel:
```
```   181   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   182   shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
```
```   183 by (erule LIM_imp_LIM, simp)
```
```   184
```
```   185 lemma LIM_norm_zero_iff:
```
```   186   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   187   shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
```
```   188 by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
```
```   189
```
```   190 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
```
```   191 by (fold real_norm_def, rule LIM_norm)
```
```   192
```
```   193 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
```
```   194 by (fold real_norm_def, rule LIM_norm_zero)
```
```   195
```
```   196 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
```
```   197 by (fold real_norm_def, rule LIM_norm_zero_cancel)
```
```   198
```
```   199 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
```
```   200 by (fold real_norm_def, rule LIM_norm_zero_iff)
```
```   201
```
```   202 lemma LIM_const_not_eq:
```
```   203   fixes a :: "'a::real_normed_algebra_1"
```
```   204   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
```
```   205 apply (simp add: LIM_def)
```
```   206 apply (rule_tac x="dist k L" in exI, simp add: zero_less_dist_iff, safe)
```
```   207 apply (rule_tac x="a + of_real (s/2)" in exI, simp add: dist_norm)
```
```   208 done
```
```   209
```
```   210 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```   211
```
```   212 lemma LIM_const_eq:
```
```   213   fixes a :: "'a::real_normed_algebra_1"
```
```   214   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
```
```   215 apply (rule ccontr)
```
```   216 apply (blast dest: LIM_const_not_eq)
```
```   217 done
```
```   218
```
```   219 lemma LIM_unique:
```
```   220   fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
```
```   221   shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
```
```   222 apply (rule ccontr)
```
```   223 apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
```
```   224 apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
```
```   225 apply (clarify, rename_tac r s)
```
```   226 apply (subgoal_tac "min r s \<noteq> 0")
```
```   227 apply (subgoal_tac "dist L M < dist L M / 2 + dist L M / 2", simp)
```
```   228 apply (subgoal_tac "dist L M \<le> dist (f (a + of_real (min r s / 2))) L +
```
```   229                                dist (f (a + of_real (min r s / 2))) M")
```
```   230 apply (erule le_less_trans, rule add_strict_mono)
```
```   231 apply (drule spec, erule mp, simp add: dist_norm)
```
```   232 apply (drule spec, erule mp, simp add: dist_norm)
```
```   233 apply (subst dist_commute, rule dist_triangle)
```
```   234 apply simp
```
```   235 done
```
```   236
```
```   237 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
```
```   238 by (auto simp add: LIM_def)
```
```   239
```
```   240 text{*Limits are equal for functions equal except at limit point*}
```
```   241 lemma LIM_equal:
```
```   242      "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
```
```   243 by (simp add: LIM_def)
```
```   244
```
```   245 lemma LIM_cong:
```
```   246   "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
```
```   247    \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
```
```   248 by (simp add: LIM_def)
```
```   249
```
```   250 lemma metric_LIM_equal2:
```
```   251   assumes 1: "0 < R"
```
```   252   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   253   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   254 apply (unfold LIM_def, safe)
```
```   255 apply (drule_tac x="r" in spec, safe)
```
```   256 apply (rule_tac x="min s R" in exI, safe)
```
```   257 apply (simp add: 1)
```
```   258 apply (simp add: 2)
```
```   259 done
```
```   260
```
```   261 lemma LIM_equal2:
```
```   262   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
```
```   263   assumes 1: "0 < R"
```
```   264   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   265   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   266 apply (unfold LIM_def dist_norm, safe)
```
```   267 apply (drule_tac x="r" in spec, safe)
```
```   268 apply (rule_tac x="min s R" in exI, safe)
```
```   269 apply (simp add: 1)
```
```   270 apply (simp add: 2)
```
```   271 done
```
```   272
```
```   273 text{*Two uses in Transcendental.ML*}
```
```   274 lemma LIM_trans:
```
```   275   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   276   shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
```
```   277 apply (drule LIM_add, assumption)
```
```   278 apply (auto simp add: add_assoc)
```
```   279 done
```
```   280
```
```   281 lemma LIM_compose:
```
```   282   assumes g: "g -- l --> g l"
```
```   283   assumes f: "f -- a --> l"
```
```   284   shows "(\<lambda>x. g (f x)) -- a --> g l"
```
```   285 proof (rule metric_LIM_I)
```
```   286   fix r::real assume r: "0 < r"
```
```   287   obtain s where s: "0 < s"
```
```   288     and less_r: "\<And>y. \<lbrakk>y \<noteq> l; dist y l < s\<rbrakk> \<Longrightarrow> dist (g y) (g l) < r"
```
```   289     using metric_LIM_D [OF g r] by fast
```
```   290   obtain t where t: "0 < t"
```
```   291     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) l < s"
```
```   292     using metric_LIM_D [OF f s] by fast
```
```   293
```
```   294   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) (g l) < r"
```
```   295   proof (rule exI, safe)
```
```   296     show "0 < t" using t .
```
```   297   next
```
```   298     fix x assume "x \<noteq> a" and "dist x a < t"
```
```   299     hence "dist (f x) l < s" by (rule less_s)
```
```   300     thus "dist (g (f x)) (g l) < r"
```
```   301       using r less_r by (case_tac "f x = l", simp_all)
```
```   302   qed
```
```   303 qed
```
```   304
```
```   305 lemma metric_LIM_compose2:
```
```   306   assumes f: "f -- a --> b"
```
```   307   assumes g: "g -- b --> c"
```
```   308   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```   309   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   310 proof (rule metric_LIM_I)
```
```   311   fix r :: real
```
```   312   assume r: "0 < r"
```
```   313   obtain s where s: "0 < s"
```
```   314     and less_r: "\<And>y. \<lbrakk>y \<noteq> b; dist y b < s\<rbrakk> \<Longrightarrow> dist (g y) c < r"
```
```   315     using metric_LIM_D [OF g r] by fast
```
```   316   obtain t where t: "0 < t"
```
```   317     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) b < s"
```
```   318     using metric_LIM_D [OF f s] by fast
```
```   319   obtain d where d: "0 < d"
```
```   320     and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"
```
```   321     using inj by fast
```
```   322
```
```   323   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) c < r"
```
```   324   proof (safe intro!: exI)
```
```   325     show "0 < min d t" using d t by simp
```
```   326   next
```
```   327     fix x
```
```   328     assume "x \<noteq> a" and "dist x a < min d t"
```
```   329     hence "f x \<noteq> b" and "dist (f x) b < s"
```
```   330       using neq_b less_s by simp_all
```
```   331     thus "dist (g (f x)) c < r"
```
```   332       by (rule less_r)
```
```   333   qed
```
```   334 qed
```
```   335
```
```   336 lemma LIM_compose2:
```
```   337   fixes a :: "'a::real_normed_vector"
```
```   338   assumes f: "f -- a --> b"
```
```   339   assumes g: "g -- b --> c"
```
```   340   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```   341   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   342 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```   343
```
```   344 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
```
```   345 unfolding o_def by (rule LIM_compose)
```
```   346
```
```   347 lemma real_LIM_sandwich_zero:
```
```   348   fixes f g :: "'a::metric_space \<Rightarrow> real"
```
```   349   assumes f: "f -- a --> 0"
```
```   350   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```   351   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```   352   shows "g -- a --> 0"
```
```   353 proof (rule LIM_imp_LIM [OF f])
```
```   354   fix x assume x: "x \<noteq> a"
```
```   355   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```   356   also have "g x \<le> f x" by (rule 2 [OF x])
```
```   357   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```   358   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```   359   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```   360 qed
```
```   361
```
```   362 text {* Bounded Linear Operators *}
```
```   363
```
```   364 lemma (in bounded_linear) LIM:
```
```   365   "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
```
```   366 unfolding LIM_conv_tendsto by (rule tendsto)
```
```   367
```
```   368 lemma (in bounded_linear) cont: "f -- a --> f a"
```
```   369 by (rule LIM [OF LIM_ident])
```
```   370
```
```   371 lemma (in bounded_linear) LIM_zero:
```
```   372   "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
```
```   373 by (drule LIM, simp only: zero)
```
```   374
```
```   375 text {* Bounded Bilinear Operators *}
```
```   376
```
```   377 lemma (in bounded_bilinear) LIM:
```
```   378   "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
```
```   379 unfolding LIM_conv_tendsto by (rule tendsto)
```
```   380
```
```   381 lemma (in bounded_bilinear) LIM_prod_zero:
```
```   382   fixes a :: "'d::metric_space"
```
```   383   assumes f: "f -- a --> 0"
```
```   384   assumes g: "g -- a --> 0"
```
```   385   shows "(\<lambda>x. f x ** g x) -- a --> 0"
```
```   386 using LIM [OF f g] by (simp add: zero_left)
```
```   387
```
```   388 lemma (in bounded_bilinear) LIM_left_zero:
```
```   389   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
```
```   390 by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
```
```   391
```
```   392 lemma (in bounded_bilinear) LIM_right_zero:
```
```   393   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
```
```   394 by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
```
```   395
```
```   396 lemmas LIM_mult = mult.LIM
```
```   397
```
```   398 lemmas LIM_mult_zero = mult.LIM_prod_zero
```
```   399
```
```   400 lemmas LIM_mult_left_zero = mult.LIM_left_zero
```
```   401
```
```   402 lemmas LIM_mult_right_zero = mult.LIM_right_zero
```
```   403
```
```   404 lemmas LIM_scaleR = scaleR.LIM
```
```   405
```
```   406 lemmas LIM_of_real = of_real.LIM
```
```   407
```
```   408 lemma LIM_power:
```
```   409   fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   410   assumes f: "f -- a --> l"
```
```   411   shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
```
```   412 by (induct n, simp, simp add: LIM_mult f)
```
```   413
```
```   414 subsubsection {* Derived theorems about @{term LIM} *}
```
```   415
```
```   416 lemma LIM_inverse_lemma:
```
```   417   fixes x :: "'a::real_normed_div_algebra"
```
```   418   assumes r: "0 < r"
```
```   419   assumes x: "norm (x - 1) < min (1/2) (r/2)"
```
```   420   shows "norm (inverse x - 1) < r"
```
```   421 proof -
```
```   422   from r have r2: "0 < r/2" by simp
```
```   423   from x have 0: "x \<noteq> 0" by clarsimp
```
```   424   from x have x': "norm (1 - x) < min (1/2) (r/2)"
```
```   425     by (simp only: norm_minus_commute)
```
```   426   hence less1: "norm (1 - x) < r/2" by simp
```
```   427   have "norm (1::'a) - norm x \<le> norm (1 - x)" by (rule norm_triangle_ineq2)
```
```   428   also from x' have "norm (1 - x) < 1/2" by simp
```
```   429   finally have "1/2 < norm x" by simp
```
```   430   hence "inverse (norm x) < inverse (1/2)"
```
```   431     by (rule less_imp_inverse_less, simp)
```
```   432   hence less2: "norm (inverse x) < 2"
```
```   433     by (simp add: nonzero_norm_inverse 0)
```
```   434   from less1 less2 r2 norm_ge_zero
```
```   435   have "norm (1 - x) * norm (inverse x) < (r/2) * 2"
```
```   436     by (rule mult_strict_mono)
```
```   437   thus "norm (inverse x - 1) < r"
```
```   438     by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0)
```
```   439 qed
```
```   440
```
```   441 lemma LIM_inverse_fun:
```
```   442   assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
```
```   443   shows "inverse -- a --> inverse a"
```
```   444 proof (rule LIM_equal2)
```
```   445   from a show "0 < norm a" by simp
```
```   446 next
```
```   447   fix x assume "norm (x - a) < norm a"
```
```   448   hence "x \<noteq> 0" by auto
```
```   449   with a show "inverse x = inverse (inverse a * x) * inverse a"
```
```   450     by (simp add: nonzero_inverse_mult_distrib
```
```   451                   nonzero_imp_inverse_nonzero
```
```   452                   nonzero_inverse_inverse_eq mult_assoc)
```
```   453 next
```
```   454   have 1: "inverse -- 1 --> inverse (1::'a)"
```
```   455     apply (rule LIM_I)
```
```   456     apply (rule_tac x="min (1/2) (r/2)" in exI)
```
```   457     apply (simp add: LIM_inverse_lemma)
```
```   458     done
```
```   459   have "(\<lambda>x. inverse a * x) -- a --> inverse a * a"
```
```   460     by (intro LIM_mult LIM_ident LIM_const)
```
```   461   hence "(\<lambda>x. inverse a * x) -- a --> 1"
```
```   462     by (simp add: a)
```
```   463   with 1 have "(\<lambda>x. inverse (inverse a * x)) -- a --> inverse 1"
```
```   464     by (rule LIM_compose)
```
```   465   hence "(\<lambda>x. inverse (inverse a * x)) -- a --> 1"
```
```   466     by simp
```
```   467   hence "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a"
```
```   468     by (intro LIM_mult LIM_const)
```
```   469   thus "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> inverse a"
```
```   470     by simp
```
```   471 qed
```
```   472
```
```   473 lemma LIM_inverse:
```
```   474   fixes L :: "'a::real_normed_div_algebra"
```
```   475   shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
```
```   476 by (rule LIM_inverse_fun [THEN LIM_compose])
```
```   477
```
```   478 lemma LIM_sgn:
```
```   479   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   480   shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
```
```   481 unfolding sgn_div_norm
```
```   482 by (simp add: LIM_scaleR LIM_inverse LIM_norm)
```
```   483
```
```   484
```
```   485 subsection {* Continuity *}
```
```   486
```
```   487 lemma LIM_isCont_iff:
```
```   488   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
```
```   489   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```   490 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```   491
```
```   492 lemma isCont_iff:
```
```   493   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
```
```   494   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```   495 by (simp add: isCont_def LIM_isCont_iff)
```
```   496
```
```   497 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
```
```   498   unfolding isCont_def by (rule LIM_ident)
```
```   499
```
```   500 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
```
```   501   unfolding isCont_def by (rule LIM_const)
```
```   502
```
```   503 lemma isCont_norm:
```
```   504   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   505   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```   506   unfolding isCont_def by (rule LIM_norm)
```
```   507
```
```   508 lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
```
```   509   unfolding isCont_def by (rule LIM_rabs)
```
```   510
```
```   511 lemma isCont_add:
```
```   512   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   513   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```   514   unfolding isCont_def by (rule LIM_add)
```
```   515
```
```   516 lemma isCont_minus:
```
```   517   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   518   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```   519   unfolding isCont_def by (rule LIM_minus)
```
```   520
```
```   521 lemma isCont_diff:
```
```   522   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   523   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```   524   unfolding isCont_def by (rule LIM_diff)
```
```   525
```
```   526 lemma isCont_mult:
```
```   527   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_algebra"
```
```   528   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```   529   unfolding isCont_def by (rule LIM_mult)
```
```   530
```
```   531 lemma isCont_inverse:
```
```   532   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   533   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
```
```   534   unfolding isCont_def by (rule LIM_inverse)
```
```   535
```
```   536 lemma isCont_LIM_compose:
```
```   537   "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
```
```   538   unfolding isCont_def by (rule LIM_compose)
```
```   539
```
```   540 lemma metric_isCont_LIM_compose2:
```
```   541   assumes f [unfolded isCont_def]: "isCont f a"
```
```   542   assumes g: "g -- f a --> l"
```
```   543   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```   544   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   545 by (rule metric_LIM_compose2 [OF f g inj])
```
```   546
```
```   547 lemma isCont_LIM_compose2:
```
```   548   fixes a :: "'a::real_normed_vector"
```
```   549   assumes f [unfolded isCont_def]: "isCont f a"
```
```   550   assumes g: "g -- f a --> l"
```
```   551   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```   552   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   553 by (rule LIM_compose2 [OF f g inj])
```
```   554
```
```   555 lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```   556   unfolding isCont_def by (rule LIM_compose)
```
```   557
```
```   558 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
```
```   559   unfolding o_def by (rule isCont_o2)
```
```   560
```
```   561 lemma (in bounded_linear) isCont: "isCont f a"
```
```   562   unfolding isCont_def by (rule cont)
```
```   563
```
```   564 lemma (in bounded_bilinear) isCont:
```
```   565   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```   566   unfolding isCont_def by (rule LIM)
```
```   567
```
```   568 lemmas isCont_scaleR = scaleR.isCont
```
```   569
```
```   570 lemma isCont_of_real:
```
```   571   "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
```
```   572   unfolding isCont_def by (rule LIM_of_real)
```
```   573
```
```   574 lemma isCont_power:
```
```   575   fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   576   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```   577   unfolding isCont_def by (rule LIM_power)
```
```   578
```
```   579 lemma isCont_sgn:
```
```   580   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   581   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
```
```   582   unfolding isCont_def by (rule LIM_sgn)
```
```   583
```
```   584 lemma isCont_abs [simp]: "isCont abs (a::real)"
```
```   585 by (rule isCont_rabs [OF isCont_ident])
```
```   586
```
```   587 lemma isCont_setsum:
```
```   588   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
```
```   589   fixes A :: "'a set" assumes "finite A"
```
```   590   shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x"
```
```   591   using `finite A`
```
```   592 proof induct
```
```   593   case (insert a F) show "isCont (\<lambda> x. \<Sum> i \<in> (insert a F). f i x) x"
```
```   594     unfolding setsum_insert[OF `finite F` `a \<notin> F`] by (rule isCont_add, auto simp add: insert)
```
```   595 qed auto
```
```   596
```
```   597 lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
```
```   598   and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
```
```   599   shows "0 \<le> f x"
```
```   600 proof (rule ccontr)
```
```   601   assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
```
```   602   hence "0 < - f x / 2" by auto
```
```   603   from isCont[unfolded isCont_def, THEN LIM_D, OF this]
```
```   604   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
```
```   605
```
```   606   let ?x = "x - min (s / 2) ((x - b) / 2)"
```
```   607   have "?x < x" and "\<bar> ?x - x \<bar> < s"
```
```   608     using `b < x` and `0 < s` by auto
```
```   609   have "b < ?x"
```
```   610   proof (cases "s < x - b")
```
```   611     case True thus ?thesis using `0 < s` by auto
```
```   612   next
```
```   613     case False hence "s / 2 \<ge> (x - b) / 2" by auto
```
```   614     from inf_absorb2[OF this, unfolded inf_real_def]
```
```   615     have "?x = (x + b) / 2" by auto
```
```   616     thus ?thesis using `b < x` by auto
```
```   617   qed
```
```   618   hence "0 \<le> f ?x" using all_le `?x < x` by auto
```
```   619   moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
```
```   620     using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
```
```   621   hence "f ?x - f x < - f x / 2" by auto
```
```   622   hence "f ?x < f x / 2" by auto
```
```   623   hence "f ?x < 0" using `f x < 0` by auto
```
```   624   thus False using `0 \<le> f ?x` by auto
```
```   625 qed
```
```   626
```
```   627
```
```   628 subsection {* Uniform Continuity *}
```
```   629
```
```   630 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```   631 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```   632
```
```   633 lemma isUCont_Cauchy:
```
```   634   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   635 unfolding isUCont_def
```
```   636 apply (rule metric_CauchyI)
```
```   637 apply (drule_tac x=e in spec, safe)
```
```   638 apply (drule_tac e=s in metric_CauchyD, safe)
```
```   639 apply (rule_tac x=M in exI, simp)
```
```   640 done
```
```   641
```
```   642 lemma (in bounded_linear) isUCont: "isUCont f"
```
```   643 unfolding isUCont_def dist_norm
```
```   644 proof (intro allI impI)
```
```   645   fix r::real assume r: "0 < r"
```
```   646   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```   647     using pos_bounded by fast
```
```   648   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```   649   proof (rule exI, safe)
```
```   650     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```   651   next
```
```   652     fix x y :: 'a
```
```   653     assume xy: "norm (x - y) < r / K"
```
```   654     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```   655     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```   656     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```   657     finally show "norm (f x - f y) < r" .
```
```   658   qed
```
```   659 qed
```
```   660
```
```   661 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   662 by (rule isUCont [THEN isUCont_Cauchy])
```
```   663
```
```   664
```
```   665 subsection {* Relation of LIM and LIMSEQ *}
```
```   666
```
```   667 lemma LIMSEQ_SEQ_conv1:
```
```   668   fixes a :: "'a::metric_space"
```
```   669   assumes X: "X -- a --> L"
```
```   670   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   671 proof (safe intro!: metric_LIMSEQ_I)
```
```   672   fix S :: "nat \<Rightarrow> 'a"
```
```   673   fix r :: real
```
```   674   assume rgz: "0 < r"
```
```   675   assume as: "\<forall>n. S n \<noteq> a"
```
```   676   assume S: "S ----> a"
```
```   677   from metric_LIM_D [OF X rgz] obtain s
```
```   678     where sgz: "0 < s"
```
```   679     and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
```
```   680     by fast
```
```   681   from metric_LIMSEQ_D [OF S sgz]
```
```   682   obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
```
```   683   hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
```
```   684   thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
```
```   685 qed
```
```   686
```
```   687
```
```   688 lemma LIMSEQ_SEQ_conv2:
```
```   689   fixes a :: real
```
```   690   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   691   shows "X -- a --> L"
```
```   692 proof (rule ccontr)
```
```   693   assume "\<not> (X -- a --> L)"
```
```   694   hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
```
```   695     unfolding LIM_def dist_norm .
```
```   696   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
```
```   697   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)
```
```   698   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
```
```   699
```
```   700   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"
```
```   701   have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
```
```   702     using rdef by simp
```
```   703   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"
```
```   704     by (rule someI_ex)
```
```   705   hence F1: "\<And>n. ?F n \<noteq> a"
```
```   706     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   707     and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
```
```   708     by fast+
```
```   709
```
```   710   have "?F ----> a"
```
```   711   proof (rule LIMSEQ_I, unfold real_norm_def)
```
```   712       fix e::real
```
```   713       assume "0 < e"
```
```   714         (* choose no such that inverse (real (Suc n)) < e *)
```
```   715       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
```
```   716       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
```
```   717       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
```
```   718       proof (intro exI allI impI)
```
```   719         fix n
```
```   720         assume mlen: "m \<le> n"
```
```   721         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   722           by (rule F2)
```
```   723         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
```
```   724           using mlen by auto
```
```   725         also from nodef have
```
```   726           "inverse (real (Suc m)) < e" .
```
```   727         finally show "\<bar>?F n - a\<bar> < e" .
```
```   728       qed
```
```   729   qed
```
```   730
```
```   731   moreover have "\<forall>n. ?F n \<noteq> a"
```
```   732     by (rule allI) (rule F1)
```
```   733
```
```   734   moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
```
```   735   ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
```
```   736
```
```   737   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
```
```   738   proof -
```
```   739     {
```
```   740       fix no::nat
```
```   741       obtain n where "n = no + 1" by simp
```
```   742       then have nolen: "no \<le> n" by simp
```
```   743         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
```
```   744       have "dist (X (?F n)) L \<ge> r"
```
```   745         by (rule F3)
```
```   746       with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
```
```   747     }
```
```   748     then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
```
```   749     with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
```
```   750     thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
```
```   751   qed
```
```   752   ultimately show False by simp
```
```   753 qed
```
```   754
```
```   755 lemma LIMSEQ_SEQ_conv:
```
```   756   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
```
```   757    (X -- a --> L)"
```
```   758 proof
```
```   759   assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   760   thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
```
```   761 next
```
```   762   assume "(X -- a --> L)"
```
```   763   thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
```
```   764 qed
```
```   765
```
```   766 end
```