src/HOL/Lim.thy
 changeset 36662 621122eeb138 parent 36661 0a5b7b818d65 child 36665 5d37a96de20c
```     1.1 --- a/src/HOL/Lim.thy	Tue May 04 10:42:47 2010 -0700
1.2 +++ b/src/HOL/Lim.thy	Tue May 04 13:08:56 2010 -0700
1.3 @@ -13,12 +13,12 @@
1.4  text{*Standard Definitions*}
1.5
1.6  abbreviation
1.7 -  LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"
1.8 +  LIM :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a, 'b] \<Rightarrow> bool"
1.9          ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
1.10    "f -- a --> L \<equiv> (f ---> L) (at a)"
1.11
1.12  definition
1.13 -  isCont :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a] \<Rightarrow> bool" where
1.14 +  isCont :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a] \<Rightarrow> bool" where
1.15    "isCont f a = (f -- a --> (f a))"
1.16
1.17  definition
1.18 @@ -61,23 +61,23 @@
1.20
1.21  lemma LIM_offset:
1.22 -  fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
1.23 +  fixes a :: "'a::real_normed_vector"
1.24    shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
1.25 -unfolding LIM_def dist_norm
1.26 -apply clarify
1.27 -apply (drule_tac x="r" in spec, safe)
1.28 -apply (rule_tac x="s" in exI, safe)
1.29 +apply (rule topological_tendstoI)
1.30 +apply (drule (2) topological_tendstoD)
1.31 +apply (simp only: eventually_at dist_norm)
1.32 +apply (clarify, rule_tac x=d in exI, safe)
1.33  apply (drule_tac x="x + k" in spec)
1.35  done
1.36
1.37  lemma LIM_offset_zero:
1.38 -  fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
1.39 +  fixes a :: "'a::real_normed_vector"
1.40    shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
1.42
1.43  lemma LIM_offset_zero_cancel:
1.44 -  fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
1.45 +  fixes a :: "'a::real_normed_vector"
1.46    shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
1.47  by (drule_tac k="- a" in LIM_offset, simp)
1.48
1.49 @@ -87,60 +87,61 @@
1.50  lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
1.51
1.53 -  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.54 +  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.55    assumes f: "f -- a --> L" and g: "g -- a --> M"
1.56    shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
1.57  using assms by (rule tendsto_add)
1.58
1.60 -  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.61 +  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.62    shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
1.63  by (drule (1) LIM_add, simp)
1.64
1.65  lemma LIM_minus:
1.66 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.67 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.68    shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
1.69  by (rule tendsto_minus)
1.70
1.71  (* TODO: delete *)
1.73 -  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.74 +  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.75    shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
1.77
1.78  lemma LIM_diff:
1.79 -  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.80 +  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.81    shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
1.82  by (rule tendsto_diff)
1.83
1.84  lemma LIM_zero:
1.85 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.86 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.87    shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
1.88 -by (simp add: LIM_def dist_norm)
1.89 +unfolding tendsto_iff dist_norm by simp
1.90
1.91  lemma LIM_zero_cancel:
1.92 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.93 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.94    shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
1.95 -by (simp add: LIM_def dist_norm)
1.96 +unfolding tendsto_iff dist_norm by simp
1.97
1.98  lemma LIM_zero_iff:
1.99    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.100    shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
1.101 -by (simp add: LIM_def dist_norm)
1.102 +unfolding tendsto_iff dist_norm by simp
1.103
1.104  lemma metric_LIM_imp_LIM:
1.105    assumes f: "f -- a --> l"
1.106    assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
1.107    shows "g -- a --> m"
1.108 -apply (rule metric_LIM_I, drule metric_LIM_D [OF f], safe)
1.109 -apply (rule_tac x="s" in exI, safe)
1.110 -apply (drule_tac x="x" in spec, safe)
1.111 +apply (rule tendstoI, drule tendstoD [OF f])
1.112 +apply (simp add: eventually_at_topological, safe)
1.113 +apply (rule_tac x="S" in exI, safe)
1.114 +apply (drule_tac x="x" in bspec, safe)
1.115  apply (erule (1) order_le_less_trans [OF le])
1.116  done
1.117
1.118  lemma LIM_imp_LIM:
1.119 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.120 -  fixes g :: "'a::metric_space \<Rightarrow> 'c::real_normed_vector"
1.121 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.122 +  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
1.123    assumes f: "f -- a --> l"
1.124    assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
1.125    shows "g -- a --> m"
1.126 @@ -149,24 +150,24 @@
1.127  done
1.128
1.129  lemma LIM_norm:
1.130 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.131 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.132    shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
1.133  by (rule tendsto_norm)
1.134
1.135  lemma LIM_norm_zero:
1.136 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.137 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.138    shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
1.139 -by (drule LIM_norm, simp)
1.140 +by (rule tendsto_norm_zero)
1.141
1.142  lemma LIM_norm_zero_cancel:
1.143 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.144 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.145    shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
1.146 -by (erule LIM_imp_LIM, simp)
1.147 +by (rule tendsto_norm_zero_cancel)
1.148
1.149  lemma LIM_norm_zero_iff:
1.150 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.151 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1.152    shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
1.153 -by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
1.154 +by (rule tendsto_norm_zero_iff)
1.155
1.156  lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
1.157  by (fold real_norm_def, rule LIM_norm)
1.158 @@ -180,40 +181,32 @@
1.159  lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
1.160  by (fold real_norm_def, rule LIM_norm_zero_iff)
1.161
1.162 +lemma at_neq_bot:
1.163 +  fixes a :: "'a::real_normed_algebra_1"
1.164 +  shows "at a \<noteq> bot"  -- {* TODO: find a more appropriate class *}
1.165 +unfolding eventually_False [symmetric]
1.166 +unfolding eventually_at dist_norm
1.167 +by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
1.168 +
1.169  lemma LIM_const_not_eq:
1.170    fixes a :: "'a::real_normed_algebra_1"
1.171 +  fixes k L :: "'b::metric_space"
1.172    shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
1.174 -apply (rule_tac x="dist k L" in exI, simp add: zero_less_dist_iff, safe)
1.175 -apply (rule_tac x="a + of_real (s/2)" in exI, simp add: dist_norm)
1.176 -done
1.177 +by (simp add: tendsto_const_iff at_neq_bot)
1.178
1.179  lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
1.180
1.181  lemma LIM_const_eq:
1.182    fixes a :: "'a::real_normed_algebra_1"
1.183 +  fixes k L :: "'b::metric_space"
1.184    shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
1.185 -apply (rule ccontr)
1.186 -apply (blast dest: LIM_const_not_eq)
1.187 -done
1.188 +by (simp add: tendsto_const_iff at_neq_bot)
1.189
1.190  lemma LIM_unique:
1.191    fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
1.192 +  fixes L M :: "'b::metric_space"
1.193    shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
1.194 -apply (rule ccontr)
1.195 -apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
1.196 -apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
1.197 -apply (clarify, rename_tac r s)
1.198 -apply (subgoal_tac "min r s \<noteq> 0")
1.199 -apply (subgoal_tac "dist L M < dist L M / 2 + dist L M / 2", simp)
1.200 -apply (subgoal_tac "dist L M \<le> dist (f (a + of_real (min r s / 2))) L +
1.201 -                               dist (f (a + of_real (min r s / 2))) M")
1.202 -apply (erule le_less_trans, rule add_strict_mono)
1.203 -apply (drule spec, erule mp, simp add: dist_norm)
1.204 -apply (drule spec, erule mp, simp add: dist_norm)
1.205 -apply (subst dist_commute, rule dist_triangle)
1.206 -apply simp
1.207 -done
1.208 +by (drule (1) tendsto_dist, simp add: tendsto_const_iff at_neq_bot)
1.209
1.210  lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
1.211  by (rule tendsto_ident_at)
1.212 @@ -221,37 +214,33 @@
1.213  text{*Limits are equal for functions equal except at limit point*}
1.214  lemma LIM_equal:
1.215       "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
1.217 +unfolding tendsto_def eventually_at_topological by simp
1.218
1.219  lemma LIM_cong:
1.220    "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
1.221     \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
1.224
1.225  lemma metric_LIM_equal2:
1.226    assumes 1: "0 < R"
1.227    assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
1.228    shows "g -- a --> l \<Longrightarrow> f -- a --> l"
1.229 -apply (unfold LIM_def, safe)
1.230 -apply (drule_tac x="r" in spec, safe)
1.231 -apply (rule_tac x="min s R" in exI, safe)
1.232 +apply (rule topological_tendstoI)
1.233 +apply (drule (2) topological_tendstoD)
1.234 +apply (simp add: eventually_at, safe)
1.235 +apply (rule_tac x="min d R" in exI, safe)
1.238  done
1.239
1.240  lemma LIM_equal2:
1.241 -  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
1.242 +  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1.243    assumes 1: "0 < R"
1.244    assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
1.245    shows "g -- a --> l \<Longrightarrow> f -- a --> l"
1.246 -apply (unfold LIM_def dist_norm, safe)
1.247 -apply (drule_tac x="r" in spec, safe)
1.248 -apply (rule_tac x="min s R" in exI, safe)
1.251 -done
1.252 +by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
1.253
1.254 -text{*Two uses in Transcendental.ML*}
1.255 +text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
1.256  lemma LIM_trans:
1.257    fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.258    shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
1.259 @@ -263,24 +252,52 @@
1.260    assumes g: "g -- l --> g l"
1.261    assumes f: "f -- a --> l"
1.262    shows "(\<lambda>x. g (f x)) -- a --> g l"
1.263 -proof (rule metric_LIM_I)
1.264 -  fix r::real assume r: "0 < r"
1.265 -  obtain s where s: "0 < s"
1.266 -    and less_r: "\<And>y. \<lbrakk>y \<noteq> l; dist y l < s\<rbrakk> \<Longrightarrow> dist (g y) (g l) < r"
1.267 -    using metric_LIM_D [OF g r] by fast
1.268 -  obtain t where t: "0 < t"
1.269 -    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) l < s"
1.270 -    using metric_LIM_D [OF f s] by fast
1.271 +proof (rule topological_tendstoI)
1.272 +  fix C assume C: "open C" "g l \<in> C"
1.273 +  obtain B where B: "open B" "l \<in> B"
1.274 +    and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> C"
1.275 +    using topological_tendstoD [OF g C]
1.276 +    unfolding eventually_at_topological by fast
1.277 +  obtain A where A: "open A" "a \<in> A"
1.278 +    and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
1.279 +    using topological_tendstoD [OF f B]
1.280 +    unfolding eventually_at_topological by fast
1.281 +  show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
1.282 +  unfolding eventually_at_topological
1.283 +  proof (intro exI conjI ballI impI)
1.284 +    show "open A" and "a \<in> A" using A .
1.285 +  next
1.286 +    fix x assume "x \<in> A" and "x \<noteq> a"
1.287 +    then show "g (f x) \<in> C"
1.288 +      by (cases "f x = l", simp add: C, simp add: gC fB)
1.289 +  qed
1.290 +qed
1.291
1.292 -  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) (g l) < r"
1.293 -  proof (rule exI, safe)
1.294 -    show "0 < t" using t .
1.295 +lemma LIM_compose_eventually:
1.296 +  assumes f: "f -- a --> b"
1.297 +  assumes g: "g -- b --> c"
1.298 +  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
1.299 +  shows "(\<lambda>x. g (f x)) -- a --> c"
1.300 +proof (rule topological_tendstoI)
1.301 +  fix C assume C: "open C" "c \<in> C"
1.302 +  obtain B where B: "open B" "b \<in> B"
1.303 +    and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> b \<Longrightarrow> g y \<in> C"
1.304 +    using topological_tendstoD [OF g C]
1.305 +    unfolding eventually_at_topological by fast
1.306 +  obtain A where A: "open A" "a \<in> A"
1.307 +    and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
1.308 +    using topological_tendstoD [OF f B]
1.309 +    unfolding eventually_at_topological by fast
1.310 +  have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> g (f x) \<in> C) (at a)"
1.311 +  unfolding eventually_at_topological
1.312 +  proof (intro exI conjI ballI impI)
1.313 +    show "open A" and "a \<in> A" using A .
1.314    next
1.315 -    fix x assume "x \<noteq> a" and "dist x a < t"
1.316 -    hence "dist (f x) l < s" by (rule less_s)
1.317 -    thus "dist (g (f x)) (g l) < r"
1.318 -      using r less_r by (case_tac "f x = l", simp_all)
1.319 +    fix x assume "x \<in> A" and "x \<noteq> a" and "f x \<noteq> b"
1.320 +    then show "g (f x) \<in> C" by (simp add: gC fB)
1.321    qed
1.322 +  with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
1.323 +    by (rule eventually_rev_mp)
1.324  qed
1.325
1.326  lemma metric_LIM_compose2:
1.327 @@ -288,31 +305,8 @@
1.328    assumes g: "g -- b --> c"
1.329    assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
1.330    shows "(\<lambda>x. g (f x)) -- a --> c"
1.331 -proof (rule metric_LIM_I)
1.332 -  fix r :: real
1.333 -  assume r: "0 < r"
1.334 -  obtain s where s: "0 < s"
1.335 -    and less_r: "\<And>y. \<lbrakk>y \<noteq> b; dist y b < s\<rbrakk> \<Longrightarrow> dist (g y) c < r"
1.336 -    using metric_LIM_D [OF g r] by fast
1.337 -  obtain t where t: "0 < t"
1.338 -    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) b < s"
1.339 -    using metric_LIM_D [OF f s] by fast
1.340 -  obtain d where d: "0 < d"
1.341 -    and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"
1.342 -    using inj by fast
1.343 -
1.344 -  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) c < r"
1.345 -  proof (safe intro!: exI)
1.346 -    show "0 < min d t" using d t by simp
1.347 -  next
1.348 -    fix x
1.349 -    assume "x \<noteq> a" and "dist x a < min d t"
1.350 -    hence "f x \<noteq> b" and "dist (f x) b < s"
1.351 -      using neq_b less_s by simp_all
1.352 -    thus "dist (g (f x)) c < r"
1.353 -      by (rule less_r)
1.354 -  qed
1.355 -qed
1.356 +using f g inj [folded eventually_at]
1.357 +by (rule LIM_compose_eventually)
1.358
1.359  lemma LIM_compose2:
1.360    fixes a :: "'a::real_normed_vector"
1.361 @@ -326,7 +320,7 @@
1.362  unfolding o_def by (rule LIM_compose)
1.363
1.364  lemma real_LIM_sandwich_zero:
1.365 -  fixes f g :: "'a::metric_space \<Rightarrow> real"
1.366 +  fixes f g :: "'a::topological_space \<Rightarrow> real"
1.367    assumes f: "f -- a --> 0"
1.368    assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
1.369    assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
1.370 @@ -593,7 +587,7 @@
1.371  subsection {* Relation of LIM and LIMSEQ *}
1.372
1.373  lemma LIMSEQ_SEQ_conv1:
1.374 -  fixes a :: "'a::metric_space"
1.375 +  fixes a :: "'a::metric_space" and L :: "'b::metric_space"
1.376    assumes X: "X -- a --> L"
1.377    shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1.378  proof (safe intro!: metric_LIMSEQ_I)
1.379 @@ -614,7 +608,7 @@
1.380
1.381
1.382  lemma LIMSEQ_SEQ_conv2:
1.383 -  fixes a :: real
1.384 +  fixes a :: real and L :: "'a::metric_space"
1.385    assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1.386    shows "X -- a --> L"
1.387  proof (rule ccontr)
1.388 @@ -682,7 +676,7 @@
1.389
1.390  lemma LIMSEQ_SEQ_conv:
1.391    "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
1.392 -   (X -- a --> L)"
1.393 +   (X -- a --> (L::'a::metric_space))"
1.394  proof
1.395    assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1.396    thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
```