src/HOL/SEQ.thy
 changeset 29197 6d4cb27ed19c parent 28952 15a4b2cf8c34 child 29667 53103fc8ffa3
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/SEQ.thy	Mon Dec 29 14:08:08 2008 +0100
1.3 @@ -0,0 +1,1136 @@
1.4 +(*  Title       : SEQ.thy
1.5 +    Author      : Jacques D. Fleuriot
1.6 +    Copyright   : 1998  University of Cambridge
1.7 +    Description : Convergence of sequences and series
1.8 +    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
1.10 +*)
1.11 +
1.12 +header {* Sequences and Convergence *}
1.13 +
1.14 +theory SEQ
1.15 +imports RealVector RComplete
1.16 +begin
1.17 +
1.18 +definition
1.19 +  Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
1.20 +    --{*Standard definition of sequence converging to zero*}
1.21 +  [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
1.22 +
1.23 +definition
1.24 +  LIMSEQ :: "[nat => 'a::real_normed_vector, 'a] => bool"
1.25 +    ("((_)/ ----> (_))" [60, 60] 60) where
1.26 +    --{*Standard definition of convergence of sequence*}
1.27 +  [code del]: "X ----> L = (\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X n - L) < r))"
1.28 +
1.29 +definition
1.30 +  lim :: "(nat => 'a::real_normed_vector) => 'a" where
1.31 +    --{*Standard definition of limit using choice operator*}
1.32 +  "lim X = (THE L. X ----> L)"
1.33 +
1.34 +definition
1.35 +  convergent :: "(nat => 'a::real_normed_vector) => bool" where
1.36 +    --{*Standard definition of convergence*}
1.37 +  "convergent X = (\<exists>L. X ----> L)"
1.38 +
1.39 +definition
1.40 +  Bseq :: "(nat => 'a::real_normed_vector) => bool" where
1.41 +    --{*Standard definition for bounded sequence*}
1.42 +  [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
1.43 +
1.44 +definition
1.45 +  monoseq :: "(nat=>real)=>bool" where
1.46 +    --{*Definition for monotonicity*}
1.47 +  [code del]: "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
1.48 +
1.49 +definition
1.50 +  subseq :: "(nat => nat) => bool" where
1.51 +    --{*Definition of subsequence*}
1.52 +  [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
1.53 +
1.54 +definition
1.55 +  Cauchy :: "(nat => 'a::real_normed_vector) => bool" where
1.56 +    --{*Standard definition of the Cauchy condition*}
1.57 +  [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. norm (X m - X n) < e)"
1.58 +
1.59 +
1.60 +subsection {* Bounded Sequences *}
1.61 +
1.62 +lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
1.63 +unfolding Bseq_def
1.64 +proof (intro exI conjI allI)
1.65 +  show "0 < max K 1" by simp
1.66 +next
1.67 +  fix n::nat
1.68 +  have "norm (X n) \<le> K" by (rule K)
1.69 +  thus "norm (X n) \<le> max K 1" by simp
1.70 +qed
1.71 +
1.72 +lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
1.73 +unfolding Bseq_def by auto
1.74 +
1.75 +lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
1.76 +proof (rule BseqI')
1.77 +  let ?A = "norm ` X ` {..N}"
1.78 +  have 1: "finite ?A" by simp
1.79 +  fix n::nat
1.80 +  show "norm (X n) \<le> max K (Max ?A)"
1.81 +  proof (cases rule: linorder_le_cases)
1.82 +    assume "n \<ge> N"
1.83 +    hence "norm (X n) \<le> K" using K by simp
1.84 +    thus "norm (X n) \<le> max K (Max ?A)" by simp
1.85 +  next
1.86 +    assume "n \<le> N"
1.87 +    hence "norm (X n) \<in> ?A" by simp
1.88 +    with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
1.89 +    thus "norm (X n) \<le> max K (Max ?A)" by simp
1.90 +  qed
1.91 +qed
1.92 +
1.93 +lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
1.94 +unfolding Bseq_def by auto
1.95 +
1.96 +lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
1.97 +apply (erule BseqE)
1.98 +apply (rule_tac N="k" and K="K" in BseqI2')
1.99 +apply clarify
1.100 +apply (drule_tac x="n - k" in spec, simp)
1.101 +done
1.102 +
1.103 +
1.104 +subsection {* Sequences That Converge to Zero *}
1.105 +
1.106 +lemma ZseqI:
1.107 +  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
1.108 +unfolding Zseq_def by simp
1.109 +
1.110 +lemma ZseqD:
1.111 +  "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
1.112 +unfolding Zseq_def by simp
1.113 +
1.114 +lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
1.115 +unfolding Zseq_def by simp
1.116 +
1.117 +lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
1.118 +unfolding Zseq_def by force
1.119 +
1.120 +lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
1.121 +unfolding Zseq_def by simp
1.122 +
1.123 +lemma Zseq_imp_Zseq:
1.124 +  assumes X: "Zseq X"
1.125 +  assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
1.126 +  shows "Zseq (\<lambda>n. Y n)"
1.127 +proof (cases)
1.128 +  assume K: "0 < K"
1.129 +  show ?thesis
1.130 +  proof (rule ZseqI)
1.131 +    fix r::real assume "0 < r"
1.132 +    hence "0 < r / K"
1.133 +      using K by (rule divide_pos_pos)
1.134 +    then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
1.135 +      using ZseqD [OF X] by fast
1.136 +    hence "\<forall>n\<ge>N. norm (X n) * K < r"
1.137 +      by (simp add: pos_less_divide_eq K)
1.138 +    hence "\<forall>n\<ge>N. norm (Y n) < r"
1.139 +      by (simp add: order_le_less_trans [OF Y])
1.140 +    thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
1.141 +  qed
1.142 +next
1.143 +  assume "\<not> 0 < K"
1.144 +  hence K: "K \<le> 0" by (simp only: linorder_not_less)
1.145 +  {
1.146 +    fix n::nat
1.147 +    have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
1.148 +    also have "\<dots> \<le> norm (X n) * 0"
1.149 +      using K norm_ge_zero by (rule mult_left_mono)
1.150 +    finally have "norm (Y n) = 0" by simp
1.151 +  }
1.152 +  thus ?thesis by (simp add: Zseq_zero)
1.153 +qed
1.154 +
1.155 +lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
1.156 +by (erule_tac K="1" in Zseq_imp_Zseq, simp)
1.157 +
1.159 +  assumes X: "Zseq X"
1.160 +  assumes Y: "Zseq Y"
1.161 +  shows "Zseq (\<lambda>n. X n + Y n)"
1.162 +proof (rule ZseqI)
1.163 +  fix r::real assume "0 < r"
1.164 +  hence r: "0 < r / 2" by simp
1.165 +  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
1.166 +    using ZseqD [OF X r] by fast
1.167 +  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
1.168 +    using ZseqD [OF Y r] by fast
1.169 +  show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
1.170 +  proof (intro exI allI impI)
1.171 +    fix n assume n: "max M N \<le> n"
1.172 +    have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
1.173 +      by (rule norm_triangle_ineq)
1.174 +    also have "\<dots> < r/2 + r/2"
1.176 +      from M n show "norm (X n) < r/2" by simp
1.177 +      from N n show "norm (Y n) < r/2" by simp
1.178 +    qed
1.179 +    finally show "norm (X n + Y n) < r" by simp
1.180 +  qed
1.181 +qed
1.182 +
1.183 +lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
1.184 +unfolding Zseq_def by simp
1.185 +
1.186 +lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
1.187 +by (simp only: diff_minus Zseq_add Zseq_minus)
1.188 +
1.189 +lemma (in bounded_linear) Zseq:
1.190 +  assumes X: "Zseq X"
1.191 +  shows "Zseq (\<lambda>n. f (X n))"
1.192 +proof -
1.193 +  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
1.194 +    using bounded by fast
1.195 +  with X show ?thesis
1.196 +    by (rule Zseq_imp_Zseq)
1.197 +qed
1.198 +
1.199 +lemma (in bounded_bilinear) Zseq:
1.200 +  assumes X: "Zseq X"
1.201 +  assumes Y: "Zseq Y"
1.202 +  shows "Zseq (\<lambda>n. X n ** Y n)"
1.203 +proof (rule ZseqI)
1.204 +  fix r::real assume r: "0 < r"
1.205 +  obtain K where K: "0 < K"
1.206 +    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
1.207 +    using pos_bounded by fast
1.208 +  from K have K': "0 < inverse K"
1.209 +    by (rule positive_imp_inverse_positive)
1.210 +  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
1.211 +    using ZseqD [OF X r] by fast
1.212 +  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
1.213 +    using ZseqD [OF Y K'] by fast
1.214 +  show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
1.215 +  proof (intro exI allI impI)
1.216 +    fix n assume n: "max M N \<le> n"
1.217 +    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
1.218 +      by (rule norm_le)
1.219 +    also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
1.220 +    proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
1.221 +      from M n show Xn: "norm (X n) < r" by simp
1.222 +      from N n show Yn: "norm (Y n) < inverse K" by simp
1.223 +    qed
1.224 +    also from K have "r * inverse K * K = r" by simp
1.225 +    finally show "norm (X n ** Y n) < r" .
1.226 +  qed
1.227 +qed
1.228 +
1.229 +lemma (in bounded_bilinear) Zseq_prod_Bseq:
1.230 +  assumes X: "Zseq X"
1.231 +  assumes Y: "Bseq Y"
1.232 +  shows "Zseq (\<lambda>n. X n ** Y n)"
1.233 +proof -
1.234 +  obtain K where K: "0 \<le> K"
1.235 +    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
1.236 +    using nonneg_bounded by fast
1.237 +  obtain B where B: "0 < B"
1.238 +    and norm_Y: "\<And>n. norm (Y n) \<le> B"
1.239 +    using Y [unfolded Bseq_def] by fast
1.240 +  from X show ?thesis
1.241 +  proof (rule Zseq_imp_Zseq)
1.242 +    fix n::nat
1.243 +    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
1.244 +      by (rule norm_le)
1.245 +    also have "\<dots> \<le> norm (X n) * B * K"
1.246 +      by (intro mult_mono' order_refl norm_Y norm_ge_zero
1.247 +                mult_nonneg_nonneg K)
1.248 +    also have "\<dots> = norm (X n) * (B * K)"
1.249 +      by (rule mult_assoc)
1.250 +    finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
1.251 +  qed
1.252 +qed
1.253 +
1.254 +lemma (in bounded_bilinear) Bseq_prod_Zseq:
1.255 +  assumes X: "Bseq X"
1.256 +  assumes Y: "Zseq Y"
1.257 +  shows "Zseq (\<lambda>n. X n ** Y n)"
1.258 +proof -
1.259 +  obtain K where K: "0 \<le> K"
1.260 +    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
1.261 +    using nonneg_bounded by fast
1.262 +  obtain B where B: "0 < B"
1.263 +    and norm_X: "\<And>n. norm (X n) \<le> B"
1.264 +    using X [unfolded Bseq_def] by fast
1.265 +  from Y show ?thesis
1.266 +  proof (rule Zseq_imp_Zseq)
1.267 +    fix n::nat
1.268 +    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
1.269 +      by (rule norm_le)
1.270 +    also have "\<dots> \<le> B * norm (Y n) * K"
1.271 +      by (intro mult_mono' order_refl norm_X norm_ge_zero
1.272 +                mult_nonneg_nonneg K)
1.273 +    also have "\<dots> = norm (Y n) * (B * K)"
1.274 +      by (simp only: mult_ac)
1.275 +    finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
1.276 +  qed
1.277 +qed
1.278 +
1.279 +lemma (in bounded_bilinear) Zseq_left:
1.280 +  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
1.281 +by (rule bounded_linear_left [THEN bounded_linear.Zseq])
1.282 +
1.283 +lemma (in bounded_bilinear) Zseq_right:
1.284 +  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
1.285 +by (rule bounded_linear_right [THEN bounded_linear.Zseq])
1.286 +
1.287 +lemmas Zseq_mult = mult.Zseq
1.288 +lemmas Zseq_mult_right = mult.Zseq_right
1.289 +lemmas Zseq_mult_left = mult.Zseq_left
1.290 +
1.291 +
1.292 +subsection {* Limits of Sequences *}
1.293 +
1.294 +lemma LIMSEQ_iff:
1.295 +      "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
1.296 +by (rule LIMSEQ_def)
1.297 +
1.298 +lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
1.299 +by (simp only: LIMSEQ_def Zseq_def)
1.300 +
1.301 +lemma LIMSEQ_I:
1.302 +  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
1.304 +
1.305 +lemma LIMSEQ_D:
1.306 +  "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
1.308 +
1.309 +lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
1.311 +
1.312 +lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l = (k = l)"
1.313 +by (simp add: LIMSEQ_Zseq_iff Zseq_const_iff)
1.314 +
1.315 +lemma LIMSEQ_norm: "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
1.316 +apply (simp add: LIMSEQ_def, safe)
1.317 +apply (drule_tac x="r" in spec, safe)
1.318 +apply (rule_tac x="no" in exI, safe)
1.319 +apply (drule_tac x="n" in spec, safe)
1.320 +apply (erule order_le_less_trans [OF norm_triangle_ineq3])
1.321 +done
1.322 +
1.323 +lemma LIMSEQ_ignore_initial_segment:
1.324 +  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
1.325 +apply (rule LIMSEQ_I)
1.326 +apply (drule (1) LIMSEQ_D)
1.327 +apply (erule exE, rename_tac N)
1.328 +apply (rule_tac x=N in exI)
1.329 +apply simp
1.330 +done
1.331 +
1.332 +lemma LIMSEQ_offset:
1.333 +  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
1.334 +apply (rule LIMSEQ_I)
1.335 +apply (drule (1) LIMSEQ_D)
1.336 +apply (erule exE, rename_tac N)
1.337 +apply (rule_tac x="N + k" in exI)
1.338 +apply clarify
1.339 +apply (drule_tac x="n - k" in spec)
1.341 +done
1.342 +
1.343 +lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
1.344 +by (drule_tac k="1" in LIMSEQ_ignore_initial_segment, simp)
1.345 +
1.346 +lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
1.347 +by (rule_tac k="1" in LIMSEQ_offset, simp)
1.348 +
1.349 +lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
1.350 +by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
1.351 +
1.353 +  fixes a b c d :: "'a::ab_group_add"
1.354 +  shows "(a + c) - (b + d) = (a - b) + (c - d)"
1.355 +by simp
1.356 +
1.357 +lemma minus_diff_minus:
1.358 +  fixes a b :: "'a::ab_group_add"
1.359 +  shows "(- a) - (- b) = - (a - b)"
1.360 +by simp
1.361 +
1.362 +lemma LIMSEQ_add: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
1.364 +
1.365 +lemma LIMSEQ_minus: "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
1.366 +by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
1.367 +
1.368 +lemma LIMSEQ_minus_cancel: "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
1.369 +by (drule LIMSEQ_minus, simp)
1.370 +
1.371 +lemma LIMSEQ_diff: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
1.373 +
1.374 +lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
1.375 +by (drule (1) LIMSEQ_diff, simp add: LIMSEQ_const_iff)
1.376 +
1.377 +lemma (in bounded_linear) LIMSEQ:
1.378 +  "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
1.379 +by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
1.380 +
1.381 +lemma (in bounded_bilinear) LIMSEQ:
1.382 +  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
1.383 +by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
1.384 +               Zseq_add Zseq Zseq_left Zseq_right)
1.385 +
1.386 +lemma LIMSEQ_mult:
1.387 +  fixes a b :: "'a::real_normed_algebra"
1.388 +  shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
1.389 +by (rule mult.LIMSEQ)
1.390 +
1.391 +lemma inverse_diff_inverse:
1.392 +  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
1.393 +   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
1.395 +
1.396 +lemma Bseq_inverse_lemma:
1.397 +  fixes x :: "'a::real_normed_div_algebra"
1.398 +  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
1.399 +apply (subst nonzero_norm_inverse, clarsimp)
1.400 +apply (erule (1) le_imp_inverse_le)
1.401 +done
1.402 +
1.403 +lemma Bseq_inverse:
1.404 +  fixes a :: "'a::real_normed_div_algebra"
1.405 +  assumes X: "X ----> a"
1.406 +  assumes a: "a \<noteq> 0"
1.407 +  shows "Bseq (\<lambda>n. inverse (X n))"
1.408 +proof -
1.409 +  from a have "0 < norm a" by simp
1.410 +  hence "\<exists>r>0. r < norm a" by (rule dense)
1.411 +  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
1.412 +  obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
1.413 +    using LIMSEQ_D [OF X r1] by fast
1.414 +  show ?thesis
1.415 +  proof (rule BseqI2' [rule_format])
1.416 +    fix n assume n: "N \<le> n"
1.417 +    hence 1: "norm (X n - a) < r" by (rule N)
1.418 +    hence 2: "X n \<noteq> 0" using r2 by auto
1.419 +    hence "norm (inverse (X n)) = inverse (norm (X n))"
1.420 +      by (rule nonzero_norm_inverse)
1.421 +    also have "\<dots> \<le> inverse (norm a - r)"
1.422 +    proof (rule le_imp_inverse_le)
1.423 +      show "0 < norm a - r" using r2 by simp
1.424 +    next
1.425 +      have "norm a - norm (X n) \<le> norm (a - X n)"
1.426 +        by (rule norm_triangle_ineq2)
1.427 +      also have "\<dots> = norm (X n - a)"
1.428 +        by (rule norm_minus_commute)
1.429 +      also have "\<dots> < r" using 1 .
1.430 +      finally show "norm a - r \<le> norm (X n)" by simp
1.431 +    qed
1.432 +    finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
1.433 +  qed
1.434 +qed
1.435 +
1.436 +lemma LIMSEQ_inverse_lemma:
1.437 +  fixes a :: "'a::real_normed_div_algebra"
1.438 +  shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
1.439 +         \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
1.440 +apply (subst LIMSEQ_Zseq_iff)
1.441 +apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
1.442 +apply (rule Zseq_minus)
1.443 +apply (rule Zseq_mult_left)
1.444 +apply (rule mult.Bseq_prod_Zseq)
1.445 +apply (erule (1) Bseq_inverse)
1.447 +done
1.448 +
1.449 +lemma LIMSEQ_inverse:
1.450 +  fixes a :: "'a::real_normed_div_algebra"
1.451 +  assumes X: "X ----> a"
1.452 +  assumes a: "a \<noteq> 0"
1.453 +  shows "(\<lambda>n. inverse (X n)) ----> inverse a"
1.454 +proof -
1.455 +  from a have "0 < norm a" by simp
1.456 +  then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
1.457 +    using LIMSEQ_D [OF X] by fast
1.458 +  hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
1.459 +  hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
1.460 +
1.461 +  from X have "(\<lambda>n. X (n + k)) ----> a"
1.462 +    by (rule LIMSEQ_ignore_initial_segment)
1.463 +  hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
1.464 +    using a k by (rule LIMSEQ_inverse_lemma)
1.465 +  thus "(\<lambda>n. inverse (X n)) ----> inverse a"
1.466 +    by (rule LIMSEQ_offset)
1.467 +qed
1.468 +
1.469 +lemma LIMSEQ_divide:
1.470 +  fixes a b :: "'a::real_normed_field"
1.471 +  shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
1.472 +by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
1.473 +
1.474 +lemma LIMSEQ_pow:
1.475 +  fixes a :: "'a::{real_normed_algebra,recpower}"
1.476 +  shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
1.477 +by (induct m) (simp_all add: power_Suc LIMSEQ_const LIMSEQ_mult)
1.478 +
1.479 +lemma LIMSEQ_setsum:
1.480 +  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
1.481 +  shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
1.482 +proof (cases "finite S")
1.483 +  case True
1.484 +  thus ?thesis using n
1.485 +  proof (induct)
1.486 +    case empty
1.487 +    show ?case
1.488 +      by (simp add: LIMSEQ_const)
1.489 +  next
1.490 +    case insert
1.491 +    thus ?case
1.493 +  qed
1.494 +next
1.495 +  case False
1.496 +  thus ?thesis
1.497 +    by (simp add: LIMSEQ_const)
1.498 +qed
1.499 +
1.500 +lemma LIMSEQ_setprod:
1.501 +  fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
1.502 +  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
1.503 +  shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
1.504 +proof (cases "finite S")
1.505 +  case True
1.506 +  thus ?thesis using n
1.507 +  proof (induct)
1.508 +    case empty
1.509 +    show ?case
1.510 +      by (simp add: LIMSEQ_const)
1.511 +  next
1.512 +    case insert
1.513 +    thus ?case
1.514 +      by (simp add: LIMSEQ_mult)
1.515 +  qed
1.516 +next
1.517 +  case False
1.518 +  thus ?thesis
1.519 +    by (simp add: setprod_def LIMSEQ_const)
1.520 +qed
1.521 +
1.522 +lemma LIMSEQ_add_const: "f ----> a ==> (%n.(f n + b)) ----> a + b"
1.524 +
1.525 +(* FIXME: delete *)
1.527 +     "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
1.528 +by (simp only: LIMSEQ_add LIMSEQ_minus)
1.529 +
1.530 +lemma LIMSEQ_diff_const: "f ----> a ==> (%n.(f n  - b)) ----> a - b"
1.531 +by (simp add: LIMSEQ_diff LIMSEQ_const)
1.532 +
1.533 +lemma LIMSEQ_diff_approach_zero:
1.534 +  "g ----> L ==> (%x. f x - g x) ----> 0  ==>
1.535 +     f ----> L"
1.537 +  apply assumption
1.538 +  apply simp
1.539 +done
1.540 +
1.541 +lemma LIMSEQ_diff_approach_zero2:
1.542 +  "f ----> L ==> (%x. f x - g x) ----> 0  ==>
1.543 +     g ----> L";
1.544 +  apply (drule LIMSEQ_diff)
1.545 +  apply assumption
1.546 +  apply simp
1.547 +done
1.548 +
1.549 +text{*A sequence tends to zero iff its abs does*}
1.550 +lemma LIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----> 0) = (X ----> 0)"
1.552 +
1.553 +lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
1.555 +
1.556 +lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
1.557 +by (drule LIMSEQ_norm, simp)
1.558 +
1.559 +text{*An unbounded sequence's inverse tends to 0*}
1.560 +
1.561 +lemma LIMSEQ_inverse_zero:
1.562 +  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
1.563 +apply (rule LIMSEQ_I)
1.564 +apply (drule_tac x="inverse r" in spec, safe)
1.565 +apply (rule_tac x="N" in exI, safe)
1.566 +apply (drule_tac x="n" in spec, safe)
1.567 +apply (frule positive_imp_inverse_positive)
1.568 +apply (frule (1) less_imp_inverse_less)
1.569 +apply (subgoal_tac "0 < X n", simp)
1.570 +apply (erule (1) order_less_trans)
1.571 +done
1.572 +
1.573 +text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
1.574 +
1.575 +lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
1.576 +apply (rule LIMSEQ_inverse_zero, safe)
1.577 +apply (cut_tac x = r in reals_Archimedean2)
1.578 +apply (safe, rule_tac x = n in exI)
1.579 +apply (auto simp add: real_of_nat_Suc)
1.580 +done
1.581 +
1.582 +text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
1.583 +infinity is now easily proved*}
1.584 +
1.586 +     "(%n. r + inverse(real(Suc n))) ----> r"
1.587 +by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
1.588 +
1.590 +     "(%n. r + -inverse(real(Suc n))) ----> r"
1.591 +by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
1.592 +
1.594 +     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
1.595 +by (cut_tac b=1 in
1.596 +        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
1.597 +
1.598 +lemma LIMSEQ_le_const:
1.599 +  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
1.600 +apply (rule ccontr, simp only: linorder_not_le)
1.601 +apply (drule_tac r="a - x" in LIMSEQ_D, simp)
1.602 +apply clarsimp
1.603 +apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
1.604 +apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
1.605 +apply simp
1.606 +done
1.607 +
1.608 +lemma LIMSEQ_le_const2:
1.609 +  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
1.610 +apply (subgoal_tac "- a \<le> - x", simp)
1.611 +apply (rule LIMSEQ_le_const)
1.612 +apply (erule LIMSEQ_minus)
1.613 +apply simp
1.614 +done
1.615 +
1.616 +lemma LIMSEQ_le:
1.617 +  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
1.618 +apply (subgoal_tac "0 \<le> y - x", simp)
1.619 +apply (rule LIMSEQ_le_const)
1.620 +apply (erule (1) LIMSEQ_diff)
1.622 +done
1.623 +
1.624 +
1.625 +subsection {* Convergence *}
1.626 +
1.627 +lemma limI: "X ----> L ==> lim X = L"
1.629 +apply (blast intro: LIMSEQ_unique)
1.630 +done
1.631 +
1.632 +lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
1.634 +
1.635 +lemma convergentI: "(X ----> L) ==> convergent X"
1.636 +by (auto simp add: convergent_def)
1.637 +
1.638 +lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
1.639 +by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
1.640 +
1.641 +lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
1.643 +apply (auto dest: LIMSEQ_minus)
1.644 +apply (drule LIMSEQ_minus, auto)
1.645 +done
1.646 +
1.647 +
1.648 +subsection {* Bounded Monotonic Sequences *}
1.649 +
1.650 +text{*Subsequence (alternative definition, (e.g. Hoskins)*}
1.651 +
1.652 +lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
1.655 +apply (induct_tac k)
1.656 +apply (auto intro: less_trans)
1.657 +done
1.658 +
1.659 +lemma monoseq_Suc:
1.660 +   "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
1.661 +                 | (\<forall>n. X (Suc n) \<le> X n))"
1.663 +apply (auto dest!: le_imp_less_or_eq)
1.664 +apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
1.665 +apply (induct_tac "ka")
1.666 +apply (auto intro: order_trans)
1.667 +apply (erule contrapos_np)
1.668 +apply (induct_tac "k")
1.669 +apply (auto intro: order_trans)
1.670 +done
1.671 +
1.672 +lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
1.674 +
1.675 +lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
1.677 +
1.678 +lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
1.680 +
1.681 +lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
1.683 +
1.684 +text{*Bounded Sequence*}
1.685 +
1.686 +lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
1.688 +
1.689 +lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
1.690 +by (auto simp add: Bseq_def)
1.691 +
1.692 +lemma lemma_NBseq_def:
1.693 +     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
1.694 +      (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
1.695 +apply auto
1.696 + prefer 2 apply force
1.697 +apply (cut_tac x = K in reals_Archimedean2, clarify)
1.698 +apply (rule_tac x = n in exI, clarify)
1.699 +apply (drule_tac x = na in spec)
1.700 +apply (auto simp add: real_of_nat_Suc)
1.701 +done
1.702 +
1.703 +text{* alternative definition for Bseq *}
1.704 +lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
1.706 +apply (simp (no_asm) add: lemma_NBseq_def)
1.707 +done
1.708 +
1.709 +lemma lemma_NBseq_def2:
1.710 +     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
1.711 +apply (subst lemma_NBseq_def, auto)
1.712 +apply (rule_tac x = "Suc N" in exI)
1.713 +apply (rule_tac [2] x = N in exI)
1.714 +apply (auto simp add: real_of_nat_Suc)
1.715 + prefer 2 apply (blast intro: order_less_imp_le)
1.716 +apply (drule_tac x = n in spec, simp)
1.717 +done
1.718 +
1.719 +(* yet another definition for Bseq *)
1.720 +lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
1.721 +by (simp add: Bseq_def lemma_NBseq_def2)
1.722 +
1.723 +subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
1.724 +
1.725 +lemma Bseq_isUb:
1.726 +  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
1.727 +by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
1.728 +
1.729 +
1.730 +text{* Use completeness of reals (supremum property)
1.731 +   to show that any bounded sequence has a least upper bound*}
1.732 +
1.733 +lemma Bseq_isLub:
1.734 +  "!!(X::nat=>real). Bseq X ==>
1.735 +   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
1.736 +by (blast intro: reals_complete Bseq_isUb)
1.737 +
1.738 +subsubsection{*A Bounded and Monotonic Sequence Converges*}
1.739 +
1.740 +lemma lemma_converg1:
1.741 +     "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
1.742 +                  isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
1.743 +               |] ==> \<forall>n \<ge> ma. X n = X ma"
1.744 +apply safe
1.745 +apply (drule_tac y = "X n" in isLubD2)
1.746 +apply (blast dest: order_antisym)+
1.747 +done
1.748 +
1.749 +text{* The best of both worlds: Easier to prove this result as a standard
1.750 +   theorem and then use equivalence to "transfer" it into the
1.751 +   equivalent nonstandard form if needed!*}
1.752 +
1.753 +lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
1.755 +apply (rule_tac x = "X m" in exI, safe)
1.756 +apply (rule_tac x = m in exI, safe)
1.757 +apply (drule spec, erule impE, auto)
1.758 +done
1.759 +
1.760 +lemma lemma_converg2:
1.761 +   "!!(X::nat=>real).
1.762 +    [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
1.763 +apply safe
1.764 +apply (drule_tac y = "X m" in isLubD2)
1.765 +apply (auto dest!: order_le_imp_less_or_eq)
1.766 +done
1.767 +
1.768 +lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
1.769 +by (rule setleI [THEN isUbI], auto)
1.770 +
1.771 +text{* FIXME: @{term "U - T < U"} is redundant *}
1.772 +lemma lemma_converg4: "!!(X::nat=> real).
1.773 +               [| \<forall>m. X m ~= U;
1.774 +                  isLub UNIV {x. \<exists>n. X n = x} U;
1.775 +                  0 < T;
1.776 +                  U + - T < U
1.777 +               |] ==> \<exists>m. U + -T < X m & X m < U"
1.778 +apply (drule lemma_converg2, assumption)
1.779 +apply (rule ccontr, simp)
1.781 +apply (drule lemma_converg3)
1.782 +apply (drule isLub_le_isUb, assumption)
1.783 +apply (auto dest: order_less_le_trans)
1.784 +done
1.785 +
1.786 +text{*A standard proof of the theorem for monotone increasing sequence*}
1.787 +
1.788 +lemma Bseq_mono_convergent:
1.789 +     "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
1.791 +apply (frule Bseq_isLub, safe)
1.792 +apply (case_tac "\<exists>m. X m = U", auto)
1.793 +apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
1.794 +(* second case *)
1.795 +apply (rule_tac x = U in exI)
1.796 +apply (subst LIMSEQ_iff, safe)
1.797 +apply (frule lemma_converg2, assumption)
1.798 +apply (drule lemma_converg4, auto)
1.799 +apply (rule_tac x = m in exI, safe)
1.800 +apply (subgoal_tac "X m \<le> X n")
1.801 + prefer 2 apply blast
1.802 +apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
1.803 +done
1.804 +
1.805 +lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
1.807 +
1.808 +text{*Main monotonicity theorem*}
1.809 +lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
1.810 +apply (simp add: monoseq_def, safe)
1.811 +apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
1.812 +apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
1.813 +apply (auto intro!: Bseq_mono_convergent)
1.814 +done
1.815 +
1.816 +subsubsection{*A Few More Equivalence Theorems for Boundedness*}
1.817 +
1.818 +text{*alternative formulation for boundedness*}
1.819 +lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
1.820 +apply (unfold Bseq_def, safe)
1.821 +apply (rule_tac [2] x = "k + norm x" in exI)
1.822 +apply (rule_tac x = K in exI, simp)
1.823 +apply (rule exI [where x = 0], auto)
1.824 +apply (erule order_less_le_trans, simp)
1.825 +apply (drule_tac x=n in spec, fold diff_def)
1.826 +apply (drule order_trans [OF norm_triangle_ineq2])
1.827 +apply simp
1.828 +done
1.829 +
1.830 +text{*alternative formulation for boundedness*}
1.831 +lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
1.832 +apply safe
1.833 +apply (simp add: Bseq_def, safe)
1.834 +apply (rule_tac x = "K + norm (X N)" in exI)
1.835 +apply auto
1.836 +apply (erule order_less_le_trans, simp)
1.837 +apply (rule_tac x = N in exI, safe)
1.838 +apply (drule_tac x = n in spec)
1.839 +apply (rule order_trans [OF norm_triangle_ineq], simp)
1.840 +apply (auto simp add: Bseq_iff2)
1.841 +done
1.842 +
1.843 +lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
1.845 +apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
1.846 +apply (drule_tac x = n in spec, arith)
1.847 +done
1.848 +
1.849 +
1.850 +subsection {* Cauchy Sequences *}
1.851 +
1.852 +lemma CauchyI:
1.853 +  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
1.855 +
1.856 +lemma CauchyD:
1.857 +  "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
1.859 +
1.860 +subsubsection {* Cauchy Sequences are Bounded *}
1.861 +
1.862 +text{*A Cauchy sequence is bounded -- this is the standard
1.863 +  proof mechanization rather than the nonstandard proof*}
1.864 +
1.865 +lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
1.866 +          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
1.867 +apply (clarify, drule spec, drule (1) mp)
1.868 +apply (simp only: norm_minus_commute)
1.869 +apply (drule order_le_less_trans [OF norm_triangle_ineq2])
1.870 +apply simp
1.871 +done
1.872 +
1.873 +lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
1.875 +apply (drule spec, drule mp, rule zero_less_one, safe)
1.876 +apply (drule_tac x="M" in spec, simp)
1.877 +apply (drule lemmaCauchy)
1.878 +apply (rule_tac k="M" in Bseq_offset)
1.880 +apply (rule_tac x="1 + norm (X M)" in exI)
1.881 +apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
1.883 +done
1.884 +
1.885 +subsubsection {* Cauchy Sequences are Convergent *}
1.886 +
1.887 +axclass banach \<subseteq> real_normed_vector
1.888 +  Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
1.889 +
1.890 +theorem LIMSEQ_imp_Cauchy:
1.891 +  assumes X: "X ----> a" shows "Cauchy X"
1.892 +proof (rule CauchyI)
1.893 +  fix e::real assume "0 < e"
1.894 +  hence "0 < e/2" by simp
1.895 +  with X have "\<exists>N. \<forall>n\<ge>N. norm (X n - a) < e/2" by (rule LIMSEQ_D)
1.896 +  then obtain N where N: "\<forall>n\<ge>N. norm (X n - a) < e/2" ..
1.897 +  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < e"
1.898 +  proof (intro exI allI impI)
1.899 +    fix m assume "N \<le> m"
1.900 +    hence m: "norm (X m - a) < e/2" using N by fast
1.901 +    fix n assume "N \<le> n"
1.902 +    hence n: "norm (X n - a) < e/2" using N by fast
1.903 +    have "norm (X m - X n) = norm ((X m - a) - (X n - a))" by simp
1.904 +    also have "\<dots> \<le> norm (X m - a) + norm (X n - a)"
1.905 +      by (rule norm_triangle_ineq4)
1.906 +    also from m n have "\<dots> < e" by(simp add:field_simps)
1.907 +    finally show "norm (X m - X n) < e" .
1.908 +  qed
1.909 +qed
1.910 +
1.911 +lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
1.912 +unfolding convergent_def
1.913 +by (erule exE, erule LIMSEQ_imp_Cauchy)
1.914 +
1.915 +text {*
1.916 +Proof that Cauchy sequences converge based on the one from
1.917 +http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
1.918 +*}
1.919 +
1.920 +text {*
1.921 +  If sequence @{term "X"} is Cauchy, then its limit is the lub of
1.922 +  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
1.923 +*}
1.924 +
1.925 +lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
1.926 +by (simp add: isUbI setleI)
1.927 +
1.928 +lemma real_abs_diff_less_iff:
1.929 +  "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
1.930 +by auto
1.931 +
1.932 +locale real_Cauchy =
1.933 +  fixes X :: "nat \<Rightarrow> real"
1.934 +  assumes X: "Cauchy X"
1.935 +  fixes S :: "real set"
1.936 +  defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
1.937 +
1.938 +lemma real_CauchyI:
1.939 +  assumes "Cauchy X"
1.940 +  shows "real_Cauchy X"
1.941 +  proof qed (fact assms)
1.942 +
1.943 +lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
1.944 +by (unfold S_def, auto)
1.945 +
1.946 +lemma (in real_Cauchy) bound_isUb:
1.947 +  assumes N: "\<forall>n\<ge>N. X n < x"
1.948 +  shows "isUb UNIV S x"
1.949 +proof (rule isUb_UNIV_I)
1.950 +  fix y::real assume "y \<in> S"
1.951 +  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
1.952 +    by (simp add: S_def)
1.953 +  then obtain M where "\<forall>n\<ge>M. y < X n" ..
1.954 +  hence "y < X (max M N)" by simp
1.955 +  also have "\<dots> < x" using N by simp
1.956 +  finally show "y \<le> x"
1.957 +    by (rule order_less_imp_le)
1.958 +qed
1.959 +
1.960 +lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
1.961 +proof (rule reals_complete)
1.962 +  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
1.963 +    using CauchyD [OF X zero_less_one] by fast
1.964 +  hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
1.965 +  show "\<exists>x. x \<in> S"
1.966 +  proof
1.967 +    from N have "\<forall>n\<ge>N. X N - 1 < X n"
1.968 +      by (simp add: real_abs_diff_less_iff)
1.969 +    thus "X N - 1 \<in> S" by (rule mem_S)
1.970 +  qed
1.971 +  show "\<exists>u. isUb UNIV S u"
1.972 +  proof
1.973 +    from N have "\<forall>n\<ge>N. X n < X N + 1"
1.974 +      by (simp add: real_abs_diff_less_iff)
1.975 +    thus "isUb UNIV S (X N + 1)"
1.976 +      by (rule bound_isUb)
1.977 +  qed
1.978 +qed
1.979 +
1.980 +lemma (in real_Cauchy) isLub_imp_LIMSEQ:
1.981 +  assumes x: "isLub UNIV S x"
1.982 +  shows "X ----> x"
1.983 +proof (rule LIMSEQ_I)
1.984 +  fix r::real assume "0 < r"
1.985 +  hence r: "0 < r/2" by simp
1.986 +  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
1.987 +    using CauchyD [OF X r] by fast
1.988 +  hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
1.989 +  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
1.990 +    by (simp only: real_norm_def real_abs_diff_less_iff)
1.991 +
1.992 +  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
1.993 +  hence "X N - r/2 \<in> S" by (rule mem_S)
1.994 +  hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
1.995 +
1.996 +  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
1.997 +  hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
1.998 +  hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
1.999 +
1.1000 +  show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
1.1001 +  proof (intro exI allI impI)
1.1002 +    fix n assume n: "N \<le> n"
1.1003 +    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
1.1004 +    thus "norm (X n - x) < r" using 1 2
1.1005 +      by (simp add: real_abs_diff_less_iff)
1.1006 +  qed
1.1007 +qed
1.1008 +
1.1009 +lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
1.1010 +proof -
1.1011 +  obtain x where "isLub UNIV S x"
1.1012 +    using isLub_ex by fast
1.1013 +  hence "X ----> x"
1.1014 +    by (rule isLub_imp_LIMSEQ)
1.1015 +  thus ?thesis ..
1.1016 +qed
1.1017 +
1.1018 +lemma real_Cauchy_convergent:
1.1019 +  fixes X :: "nat \<Rightarrow> real"
1.1020 +  shows "Cauchy X \<Longrightarrow> convergent X"
1.1021 +unfolding convergent_def
1.1022 +by (rule real_Cauchy.LIMSEQ_ex)
1.1023 + (rule real_CauchyI)
1.1024 +
1.1025 +instance real :: banach
1.1026 +by intro_classes (rule real_Cauchy_convergent)
1.1027 +
1.1028 +lemma Cauchy_convergent_iff:
1.1029 +  fixes X :: "nat \<Rightarrow> 'a::banach"
1.1030 +  shows "Cauchy X = convergent X"
1.1031 +by (fast intro: Cauchy_convergent convergent_Cauchy)
1.1032 +
1.1033 +
1.1034 +subsection {* Power Sequences *}
1.1035 +
1.1036 +text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
1.1037 +"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
1.1038 +  also fact that bounded and monotonic sequence converges.*}
1.1039 +
1.1040 +lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
1.1042 +apply (rule_tac x = 1 in exI)
1.1044 +apply (auto dest: power_mono)
1.1045 +done
1.1046 +
1.1047 +lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
1.1048 +apply (clarify intro!: mono_SucI2)
1.1049 +apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
1.1050 +done
1.1051 +
1.1052 +lemma convergent_realpow:
1.1053 +  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
1.1054 +by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
1.1055 +
1.1056 +lemma LIMSEQ_inverse_realpow_zero_lemma:
1.1057 +  fixes x :: real
1.1058 +  assumes x: "0 \<le> x"
1.1059 +  shows "real n * x + 1 \<le> (x + 1) ^ n"
1.1060 +apply (induct n)
1.1061 +apply simp
1.1062 +apply simp
1.1063 +apply (rule order_trans)
1.1064 +prefer 2
1.1065 +apply (erule mult_left_mono)
1.1066 +apply (rule add_increasing [OF x], simp)
1.1069 +apply (simp add: mult_nonneg_nonneg x)
1.1070 +done
1.1071 +
1.1072 +lemma LIMSEQ_inverse_realpow_zero:
1.1073 +  "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
1.1074 +proof (rule LIMSEQ_inverse_zero [rule_format])
1.1075 +  fix y :: real
1.1076 +  assume x: "1 < x"
1.1077 +  hence "0 < x - 1" by simp
1.1078 +  hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
1.1079 +    by (rule reals_Archimedean3)
1.1080 +  hence "\<exists>N::nat. y < real N * (x - 1)" ..
1.1081 +  then obtain N::nat where "y < real N * (x - 1)" ..
1.1082 +  also have "\<dots> \<le> real N * (x - 1) + 1" by simp
1.1083 +  also have "\<dots> \<le> (x - 1 + 1) ^ N"
1.1084 +    by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
1.1085 +  also have "\<dots> = x ^ N" by simp
1.1086 +  finally have "y < x ^ N" .
1.1087 +  hence "\<forall>n\<ge>N. y < x ^ n"
1.1088 +    apply clarify
1.1089 +    apply (erule order_less_le_trans)
1.1090 +    apply (erule power_increasing)
1.1091 +    apply (rule order_less_imp_le [OF x])
1.1092 +    done
1.1093 +  thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
1.1094 +qed
1.1095 +
1.1096 +lemma LIMSEQ_realpow_zero:
1.1097 +  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1.1098 +proof (cases)
1.1099 +  assume "x = 0"
1.1100 +  hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
1.1101 +  thus ?thesis by (rule LIMSEQ_imp_Suc)
1.1102 +next
1.1103 +  assume "0 \<le> x" and "x \<noteq> 0"
1.1104 +  hence x0: "0 < x" by simp
1.1105 +  assume x1: "x < 1"
1.1106 +  from x0 x1 have "1 < inverse x"
1.1107 +    by (rule real_inverse_gt_one)
1.1108 +  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
1.1109 +    by (rule LIMSEQ_inverse_realpow_zero)
1.1110 +  thus ?thesis by (simp add: power_inverse)
1.1111 +qed
1.1112 +
1.1113 +lemma LIMSEQ_power_zero:
1.1114 +  fixes x :: "'a::{real_normed_algebra_1,recpower}"
1.1115 +  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1.1116 +apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
1.1117 +apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
1.1118 +apply (simp add: power_abs norm_power_ineq)
1.1119 +done
1.1120 +
1.1121 +lemma LIMSEQ_divide_realpow_zero:
1.1122 +  "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
1.1123 +apply (cut_tac a = a and x1 = "inverse x" in
1.1124 +        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
1.1125 +apply (auto simp add: divide_inverse power_inverse)
1.1126 +apply (simp add: inverse_eq_divide pos_divide_less_eq)
1.1127 +done
1.1128 +
1.1129 +text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
1.1130 +
1.1131 +lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
1.1132 +by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
1.1133 +
1.1134 +lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
1.1135 +apply (rule LIMSEQ_rabs_zero [THEN iffD1])
1.1136 +apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
1.1137 +done
1.1138 +
1.1139 +end
```