src/HOL/SEQ.thy
 author huffman Sun, 31 May 2009 21:59:33 -0700 changeset 31349 2261c8781f73 parent 31336 e17f13cd1280 child 31353 14a58e2ca374 permissions -rw-r--r--
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
```
(*  Title       : SEQ.thy
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
Description : Convergence of sequences and series
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

header {* Sequences and Convergence *}

theory SEQ
imports Limits
begin

definition
sequentially :: "nat filter" where
"sequentially = Abs_filter (\<lambda>P. \<exists>N. \<forall>n\<ge>N. P n)"

definition
Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
--{*Standard definition of sequence converging to zero*}
[code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"

definition
LIMSEQ :: "[nat \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
("((_)/ ----> (_))" [60, 60] 60) where
--{*Standard definition of convergence of sequence*}
[code del]: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"

definition
lim :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where
--{*Standard definition of limit using choice operator*}
"lim X = (THE L. X ----> L)"

definition
convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
--{*Standard definition of convergence*}
"convergent X = (\<exists>L. X ----> L)"

definition
Bseq :: "(nat => 'a::real_normed_vector) => bool" where
--{*Standard definition for bounded sequence*}
[code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"

definition
monoseq :: "(nat=>real)=>bool" where
--{*Definition of monotonicity.
The use of disjunction here complicates proofs considerably.
One alternative is to add a Boolean argument to indicate the direction.
Another is to develop the notions of increasing and decreasing first.*}
[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))"

definition
incseq :: "(nat=>real)=>bool" where
--{*Increasing sequence*}
[code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"

definition
decseq :: "(nat=>real)=>bool" where
--{*Increasing sequence*}
[code del]: "decseq X = (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"

definition
subseq :: "(nat => nat) => bool" where
--{*Definition of subsequence*}
[code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"

definition
Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
--{*Standard definition of the Cauchy condition*}
[code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"

subsection {* Sequentially *}

lemma eventually_sequentially:
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
unfolding sequentially_def
apply (rule eventually_Abs_filter)
apply simp
apply (clarify, rule_tac x=N in exI, simp)
apply (clarify, rename_tac M N)
apply (rule_tac x="max M N" in exI, simp)
done

lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
unfolding Zseq_def Zfun_def eventually_sequentially ..

lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
unfolding LIMSEQ_def tendsto_def eventually_sequentially ..

subsection {* Bounded Sequences *}

lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
unfolding Bseq_def
proof (intro exI conjI allI)
show "0 < max K 1" by simp
next
fix n::nat
have "norm (X n) \<le> K" by (rule K)
thus "norm (X n) \<le> max K 1" by simp
qed

lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Bseq_def by auto

lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
proof (rule BseqI')
let ?A = "norm ` X ` {..N}"
have 1: "finite ?A" by simp
fix n::nat
show "norm (X n) \<le> max K (Max ?A)"
proof (cases rule: linorder_le_cases)
assume "n \<ge> N"
hence "norm (X n) \<le> K" using K by simp
thus "norm (X n) \<le> max K (Max ?A)" by simp
next
assume "n \<le> N"
hence "norm (X n) \<in> ?A" by simp
with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
thus "norm (X n) \<le> max K (Max ?A)" by simp
qed
qed

lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
unfolding Bseq_def by auto

lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
apply (erule BseqE)
apply (rule_tac N="k" and K="K" in BseqI2')
apply clarify
apply (drule_tac x="n - k" in spec, simp)
done

subsection {* Sequences That Converge to Zero *}

lemma ZseqI:
"(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
unfolding Zseq_def by simp

lemma ZseqD:
"\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
unfolding Zseq_def by simp

lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
unfolding Zseq_def by simp

lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
unfolding Zseq_def by force

lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
unfolding Zseq_def by simp

lemma Zseq_imp_Zseq:
assumes X: "Zseq X"
assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
shows "Zseq (\<lambda>n. Y n)"
using assms unfolding Zseq_conv_Zfun by (rule Zfun_imp_Zfun)

lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
by (erule_tac K="1" in Zseq_imp_Zseq, simp)

"Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"

lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
unfolding Zseq_def by simp

lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
by (simp only: diff_minus Zseq_add Zseq_minus)

lemma (in bounded_linear) Zseq:
"Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
unfolding Zseq_conv_Zfun by (rule Zfun)

lemma (in bounded_bilinear) Zseq:
"Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
unfolding Zseq_conv_Zfun by (rule Zfun)

lemma (in bounded_bilinear) Zseq_prod_Bseq:
assumes X: "Zseq X"
assumes Y: "Bseq Y"
shows "Zseq (\<lambda>n. X n ** Y n)"
proof -
obtain K where K: "0 \<le> K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
using nonneg_bounded by fast
obtain B where B: "0 < B"
and norm_Y: "\<And>n. norm (Y n) \<le> B"
using Y [unfolded Bseq_def] by fast
from X show ?thesis
proof (rule Zseq_imp_Zseq)
fix n::nat
have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
by (rule norm_le)
also have "\<dots> \<le> norm (X n) * B * K"
by (intro mult_mono' order_refl norm_Y norm_ge_zero
mult_nonneg_nonneg K)
also have "\<dots> = norm (X n) * (B * K)"
by (rule mult_assoc)
finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
qed
qed

lemma (in bounded_bilinear) Bseq_prod_Zseq:
assumes X: "Bseq X"
assumes Y: "Zseq Y"
shows "Zseq (\<lambda>n. X n ** Y n)"
proof -
obtain K where K: "0 \<le> K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
using nonneg_bounded by fast
obtain B where B: "0 < B"
and norm_X: "\<And>n. norm (X n) \<le> B"
using X [unfolded Bseq_def] by fast
from Y show ?thesis
proof (rule Zseq_imp_Zseq)
fix n::nat
have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
by (rule norm_le)
also have "\<dots> \<le> B * norm (Y n) * K"
by (intro mult_mono' order_refl norm_X norm_ge_zero
mult_nonneg_nonneg K)
also have "\<dots> = norm (Y n) * (B * K)"
by (simp only: mult_ac)
finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
qed
qed

lemma (in bounded_bilinear) Zseq_left:
"Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
by (rule bounded_linear_left [THEN bounded_linear.Zseq])

lemma (in bounded_bilinear) Zseq_right:
"Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
by (rule bounded_linear_right [THEN bounded_linear.Zseq])

lemmas Zseq_mult = mult.Zseq
lemmas Zseq_mult_right = mult.Zseq_right
lemmas Zseq_mult_left = mult.Zseq_left

subsection {* Limits of Sequences *}

lemma LIMSEQ_iff:
fixes L :: "'a::real_normed_vector"
shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
unfolding LIMSEQ_def dist_norm ..

lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
by (simp only: LIMSEQ_iff Zseq_def)

lemma metric_LIMSEQ_I:
"(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"

lemma metric_LIMSEQ_D:
"\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"

lemma LIMSEQ_I:
fixes L :: "'a::real_normed_vector"
shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"

lemma LIMSEQ_D:
fixes L :: "'a::real_normed_vector"
shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"

lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"

lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
apply (safe intro!: LIMSEQ_const)
apply (rule ccontr)
apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
apply auto
done

lemma LIMSEQ_norm:
fixes a :: "'a::real_normed_vector"
shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)

lemma LIMSEQ_ignore_initial_segment:
"f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
apply (rule metric_LIMSEQ_I)
apply (drule (1) metric_LIMSEQ_D)
apply (erule exE, rename_tac N)
apply (rule_tac x=N in exI)
apply simp
done

lemma LIMSEQ_offset:
"(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
apply (rule metric_LIMSEQ_I)
apply (drule (1) metric_LIMSEQ_D)
apply (erule exE, rename_tac N)
apply (rule_tac x="N + k" in exI)
apply clarify
apply (drule_tac x="n - k" in spec)
done

lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)

lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)

lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)

lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
unfolding LIMSEQ_def
by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)

fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"

lemma LIMSEQ_minus:
fixes a :: "'a::real_normed_vector"
shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)

lemma LIMSEQ_minus_cancel:
fixes a :: "'a::real_normed_vector"
shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
by (drule LIMSEQ_minus, simp)

lemma LIMSEQ_diff:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)

lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
apply (rule ccontr)
apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
apply (clarify, rename_tac M N)
apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
apply (erule le_less_trans, rule add_strict_mono, simp, simp)
apply (subst dist_commute, rule dist_triangle)
done

lemma (in bounded_linear) LIMSEQ:
"X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
unfolding LIMSEQ_conv_tendsto by (rule tendsto)

lemma (in bounded_bilinear) LIMSEQ:
"\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
unfolding LIMSEQ_conv_tendsto by (rule tendsto)

lemma LIMSEQ_mult:
fixes a b :: "'a::real_normed_algebra"
shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
by (rule mult.LIMSEQ)

lemma inverse_diff_inverse:
"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
\<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"

lemma Bseq_inverse_lemma:
fixes x :: "'a::real_normed_div_algebra"
shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
apply (subst nonzero_norm_inverse, clarsimp)
apply (erule (1) le_imp_inverse_le)
done

lemma Bseq_inverse:
fixes a :: "'a::real_normed_div_algebra"
assumes X: "X ----> a"
assumes a: "a \<noteq> 0"
shows "Bseq (\<lambda>n. inverse (X n))"
proof -
from a have "0 < norm a" by simp
hence "\<exists>r>0. r < norm a" by (rule dense)
then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
using LIMSEQ_D [OF X r1] by fast
show ?thesis
proof (rule BseqI2' [rule_format])
fix n assume n: "N \<le> n"
hence 1: "norm (X n - a) < r" by (rule N)
hence 2: "X n \<noteq> 0" using r2 by auto
hence "norm (inverse (X n)) = inverse (norm (X n))"
by (rule nonzero_norm_inverse)
also have "\<dots> \<le> inverse (norm a - r)"
proof (rule le_imp_inverse_le)
show "0 < norm a - r" using r2 by simp
next
have "norm a - norm (X n) \<le> norm (a - X n)"
by (rule norm_triangle_ineq2)
also have "\<dots> = norm (X n - a)"
by (rule norm_minus_commute)
also have "\<dots> < r" using 1 .
finally show "norm a - r \<le> norm (X n)" by simp
qed
finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
qed
qed

lemma LIMSEQ_inverse_lemma:
fixes a :: "'a::real_normed_div_algebra"
shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
\<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
apply (subst LIMSEQ_Zseq_iff)
apply (rule Zseq_minus)
apply (rule Zseq_mult_left)
apply (rule mult.Bseq_prod_Zseq)
apply (erule (1) Bseq_inverse)
done

lemma LIMSEQ_inverse:
fixes a :: "'a::real_normed_div_algebra"
assumes X: "X ----> a"
assumes a: "a \<noteq> 0"
shows "(\<lambda>n. inverse (X n)) ----> inverse a"
proof -
from a have "0 < norm a" by simp
then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
using LIMSEQ_D [OF X] by fast
hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp

from X have "(\<lambda>n. X (n + k)) ----> a"
by (rule LIMSEQ_ignore_initial_segment)
hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
using a k by (rule LIMSEQ_inverse_lemma)
thus "(\<lambda>n. inverse (X n)) ----> inverse a"
by (rule LIMSEQ_offset)
qed

lemma LIMSEQ_divide:
fixes a b :: "'a::real_normed_field"
shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)

lemma LIMSEQ_pow:
fixes a :: "'a::{power, real_normed_algebra}"
shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)

lemma LIMSEQ_setsum:
fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
proof (cases "finite S")
case True
thus ?thesis using n
proof (induct)
case empty
show ?case
next
case insert
thus ?case
qed
next
case False
thus ?thesis
qed

lemma LIMSEQ_setprod:
fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
proof (cases "finite S")
case True
thus ?thesis using n
proof (induct)
case empty
show ?case
next
case insert
thus ?case
qed
next
case False
thus ?thesis
qed

fixes a :: "'a::real_normed_vector"
shows "f ----> a ==> (%n.(f n + b)) ----> a + b"

(* FIXME: delete *)
fixes a b :: "'a::real_normed_vector"
shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"

lemma LIMSEQ_diff_const:
fixes a b :: "'a::real_normed_vector"
shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"

lemma LIMSEQ_diff_approach_zero:
fixes L :: "'a::real_normed_vector"
shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"

lemma LIMSEQ_diff_approach_zero2:
fixes L :: "'a::real_normed_vector"
shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L";
by (drule (1) LIMSEQ_diff, simp)

text{*A sequence tends to zero iff its abs does*}
lemma LIMSEQ_norm_zero:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"

lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"

lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
by (drule LIMSEQ_norm, simp)

text{*An unbounded sequence's inverse tends to 0*}

lemma LIMSEQ_inverse_zero:
"\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
apply (rule LIMSEQ_I)
apply (drule_tac x="inverse r" in spec, safe)
apply (rule_tac x="N" in exI, safe)
apply (drule_tac x="n" in spec, safe)
apply (frule positive_imp_inverse_positive)
apply (frule (1) less_imp_inverse_less)
apply (subgoal_tac "0 < X n", simp)
apply (erule (1) order_less_trans)
done

text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}

lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
apply (rule LIMSEQ_inverse_zero, safe)
apply (cut_tac x = r in reals_Archimedean2)
apply (safe, rule_tac x = n in exI)
done

text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
infinity is now easily proved*}

"(%n. r + inverse(real(Suc n))) ----> r"
by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)

"(%n. r + -inverse(real(Suc n))) ----> r"
by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)

"(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
by (cut_tac b=1 in

lemma LIMSEQ_le_const:
"\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
apply (rule ccontr, simp only: linorder_not_le)
apply (drule_tac r="a - x" in LIMSEQ_D, simp)
apply clarsimp
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
apply simp
done

lemma LIMSEQ_le_const2:
"\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
apply (subgoal_tac "- a \<le> - x", simp)
apply (rule LIMSEQ_le_const)
apply (erule LIMSEQ_minus)
apply simp
done

lemma LIMSEQ_le:
"\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
apply (subgoal_tac "0 \<le> y - x", simp)
apply (rule LIMSEQ_le_const)
apply (erule (1) LIMSEQ_diff)
done

subsection {* Convergence *}

lemma limI: "X ----> L ==> lim X = L"
apply (blast intro: LIMSEQ_unique)
done

lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"

lemma convergentI: "(X ----> L) ==> convergent X"

lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)

lemma convergent_minus_iff:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
apply (auto dest: LIMSEQ_minus)
apply (drule LIMSEQ_minus, auto)
done

text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}

lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
unfolding Ex1_def
apply (rule_tac x="nat_rec e f" in exI)
apply (rule conjI)+
apply (rule def_nat_rec_0, simp)
apply (rule allI, rule def_nat_rec_Suc, simp)
apply (rule allI, rule impI, rule ext)
apply (erule conjE)
apply (induct_tac x)
apply (erule_tac x="n" in allE)
apply (simp)
done

text{*Subsequence (alternative definition, (e.g. Hoskins)*}

lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
apply (induct_tac k)
apply (auto intro: less_trans)
done

lemma monoseq_Suc:
"monoseq X = ((\<forall>n. X n \<le> X (Suc n))
| (\<forall>n. X (Suc n) \<le> X n))"
apply (auto dest!: le_imp_less_or_eq)
apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
apply (induct_tac "ka")
apply (auto intro: order_trans)
apply (erule contrapos_np)
apply (induct_tac "k")
apply (auto intro: order_trans)
done

lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"

lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"

lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"

lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"

lemma monoseq_minus: assumes "monoseq a"
shows "monoseq (\<lambda> n. - a n)"
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
case True
hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
thus ?thesis by (rule monoI2)
next
case False
hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
thus ?thesis by (rule monoI1)
qed

lemma monoseq_le: assumes "monoseq a" and "a ----> x"
shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
proof -
{ fix x n fix a :: "nat \<Rightarrow> real"
assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
have "a n \<le> x"
proof (rule ccontr)
assume "\<not> a n \<le> x" hence "x < a n" by auto
hence "0 < a n - x" by auto
from `a ----> x`[THEN LIMSEQ_D, OF this]
obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
hence "norm (a (max no n) - x) < a n - x" by auto
moreover
{ fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
hence "x < a (max no n)" by auto
ultimately
have "a (max no n) < a n" by auto
with monotone[where m=n and n="max no n"]
show False by auto
qed
} note top_down = this
{ fix x n m fix a :: "nat \<Rightarrow> real"
assume "a ----> x" and "monoseq a" and "a m < x"
have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
case True with top_down and `a ----> x` show ?thesis by auto
next
case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
hence False using `a m < x` by auto
thus ?thesis ..
qed
} note when_decided = this

show ?thesis
proof (cases "\<exists> m. a m \<noteq> x")
case True then obtain m where "a m \<noteq> x" by auto
show ?thesis
proof (cases "a m < x")
case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
show ?thesis by blast
next
case False hence "- a m < - x" using `a m \<noteq> x` by auto
with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
show ?thesis by auto
qed
qed auto
qed

text{* for any sequence, there is a mootonic subsequence *}
lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
proof-
{assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
using H apply -
apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
unfolding order_le_less by blast
hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
{fix n
have "?P (f (Suc n)) (f n)"
unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
using H apply -
apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
unfolding order_le_less by blast
hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
note fSuc = this
{fix p q assume pq: "p \<ge> f q"
have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
by (cases q, simp_all) }
note pqth = this
{fix q
have "f (Suc q) > f q" apply (induct q)
using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
note fss = this
from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
{fix a b
have "f a \<le> f (a + b)"
proof(induct b)
case 0 thus ?case by simp
next
case (Suc b)
from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
qed}
note fmon0 = this
have "monoseq (\<lambda>n. s (f n))"
proof-
{fix n
have "s (f n) \<ge> s (f (Suc n))"
proof(cases n)
case 0
assume n0: "n = 0"
from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
next
case (Suc m)
assume m: "n = Suc m"
from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
qed}
thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
qed
with th1 have ?thesis by blast}
moreover
{fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
{fix p assume p: "p \<ge> Suc N"
hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
have "m \<noteq> p" using m(2) by auto
with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
note th0 = this
let ?P = "\<lambda>m x. m > x \<and> s x < s m"
from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
"\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
using N apply -
apply (erule allE[where x="Suc N"], clarsimp)
apply (rule_tac x="m" in exI)
apply auto
apply (subgoal_tac "Suc N \<noteq> m")
apply simp
apply (rule ccontr, simp)
done
hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
{fix n
have "f n > N \<and> ?P (f (Suc n)) (f n)"
unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
proof (induct n)
case 0 thus ?case
using f0 N apply auto
apply (erule allE[where x="f 0"], clarsimp)
apply (rule_tac x="m" in exI, simp)
by (subgoal_tac "f 0 \<noteq> m", auto)
next
case (Suc n)
from Suc.hyps have Nfn: "N < f n" by blast
from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
with Nfn have mN: "m > N" by arith
note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]

from key have th0: "f (Suc n) > N" by simp
from N[rule_format, OF th0]
obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
hence "m' > f (Suc n)" using m'(1) by simp
with key m'(2) show ?case by auto
qed}
note fSuc = this
{fix n
have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto
hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
note thf = this
have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
apply -
apply (rule disjI1)
apply auto
apply (rule order_less_imp_le)
apply blast
done
then have ?thesis  using sqf by blast}
ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
qed

lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
proof(induct n)
case 0 thus ?case by simp
next
case (Suc n)
from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
have "n < f (Suc n)" by arith
thus ?case by arith
qed

lemma LIMSEQ_subseq_LIMSEQ:
"\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
apply (drule_tac x=r in spec, clarify)
apply (rule_tac x=no in exI, clarify)
apply (blast intro: seq_suble le_trans dest!: spec)
done

subsection {* Bounded Monotonic Sequences *}

text{*Bounded Sequence*}

lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"

lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"

lemma lemma_NBseq_def:
"(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
(\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
apply auto
prefer 2 apply force
apply (cut_tac x = K in reals_Archimedean2, clarify)
apply (rule_tac x = n in exI, clarify)
apply (drule_tac x = na in spec)
done

text{* alternative definition for Bseq *}
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
done

lemma lemma_NBseq_def2:
"(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
apply (subst lemma_NBseq_def, auto)
apply (rule_tac x = "Suc N" in exI)
apply (rule_tac [2] x = N in exI)
prefer 2 apply (blast intro: order_less_imp_le)
apply (drule_tac x = n in spec, simp)
done

(* yet another definition for Bseq *)
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"

subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}

lemma Bseq_isUb:
"!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)

text{* Use completeness of reals (supremum property)
to show that any bounded sequence has a least upper bound*}

lemma Bseq_isLub:
"!!(X::nat=>real). Bseq X ==>
\<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
by (blast intro: reals_complete Bseq_isUb)

subsubsection{*A Bounded and Monotonic Sequence Converges*}

lemma lemma_converg1:
"!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
|] ==> \<forall>n \<ge> ma. X n = X ma"
apply safe
apply (drule_tac y = "X n" in isLubD2)
apply (blast dest: order_antisym)+
done

text{* The best of both worlds: Easier to prove this result as a standard
theorem and then use equivalence to "transfer" it into the
equivalent nonstandard form if needed!*}

lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
apply (rule_tac x = "X m" in exI, safe)
apply (rule_tac x = m in exI, safe)
apply (drule spec, erule impE, auto)
done

lemma lemma_converg2:
"!!(X::nat=>real).
[| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
apply safe
apply (drule_tac y = "X m" in isLubD2)
apply (auto dest!: order_le_imp_less_or_eq)
done

lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
by (rule setleI [THEN isUbI], auto)

text{* FIXME: @{term "U - T < U"} is redundant *}
lemma lemma_converg4: "!!(X::nat=> real).
[| \<forall>m. X m ~= U;
isLub UNIV {x. \<exists>n. X n = x} U;
0 < T;
U + - T < U
|] ==> \<exists>m. U + -T < X m & X m < U"
apply (drule lemma_converg2, assumption)
apply (rule ccontr, simp)
apply (drule lemma_converg3)
apply (drule isLub_le_isUb, assumption)
apply (auto dest: order_less_le_trans)
done

text{*A standard proof of the theorem for monotone increasing sequence*}

lemma Bseq_mono_convergent:
"[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
apply (frule Bseq_isLub, safe)
apply (case_tac "\<exists>m. X m = U", auto)
apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
(* second case *)
apply (rule_tac x = U in exI)
apply (subst LIMSEQ_iff, safe)
apply (frule lemma_converg2, assumption)
apply (drule lemma_converg4, auto)
apply (rule_tac x = m in exI, safe)
apply (subgoal_tac "X m \<le> X n")
prefer 2 apply blast
apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
done

lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"

text{*Main monotonicity theorem*}
lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
apply (auto intro!: Bseq_mono_convergent)
done

subsubsection{*Increasing and Decreasing Series*}

lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"

lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
proof
assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
thus ?thesis by simp
next
assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
by (auto simp add: incseq_def intro: order_antisym)
have X: "!!n. X n = X 0"
by (blast intro: const [of 0])
have "X = (\<lambda>n. X 0)"
by (blast intro: ext X)
hence "L = X 0" using LIMSEQ_const [of "X 0"]
by (auto intro: LIMSEQ_unique lim)
thus ?thesis
by (blast intro: eq_refl X)
qed

lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"

lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"

lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
proof -
have inc: "incseq (\<lambda>n. - X n)" using dec
have "- X n \<le> - L"
by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim)
thus ?thesis
by simp
qed

subsubsection{*A Few More Equivalence Theorems for Boundedness*}

text{*alternative formulation for boundedness*}
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
apply (unfold Bseq_def, safe)
apply (rule_tac [2] x = "k + norm x" in exI)
apply (rule_tac x = K in exI, simp)
apply (rule exI [where x = 0], auto)
apply (erule order_less_le_trans, simp)
apply (drule_tac x=n in spec, fold diff_def)
apply (drule order_trans [OF norm_triangle_ineq2])
apply simp
done

text{*alternative formulation for boundedness*}
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
apply safe
apply (rule_tac x = "K + norm (X N)" in exI)
apply auto
apply (erule order_less_le_trans, simp)
apply (rule_tac x = N in exI, safe)
apply (drule_tac x = n in spec)
apply (rule order_trans [OF norm_triangle_ineq], simp)
done

lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
apply (drule_tac x = n in spec, arith)
done

subsection {* Cauchy Sequences *}

lemma metric_CauchyI:
"(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"

lemma metric_CauchyD:
"\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"

lemma Cauchy_iff:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
unfolding Cauchy_def dist_norm ..

lemma CauchyI:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"

lemma CauchyD:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"

lemma Cauchy_subseq_Cauchy:
"\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
apply (drule_tac x=e in spec, clarify)
apply (rule_tac x=M in exI, clarify)
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
done

subsubsection {* Cauchy Sequences are Bounded *}

text{*A Cauchy sequence is bounded -- this is the standard
proof mechanization rather than the nonstandard proof*}

lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
apply (clarify, drule spec, drule (1) mp)
apply (simp only: norm_minus_commute)
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
apply simp
done

lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
apply (drule spec, drule mp, rule zero_less_one, safe)
apply (drule_tac x="M" in spec, simp)
apply (drule lemmaCauchy)
apply (rule_tac k="M" in Bseq_offset)
apply (rule_tac x="1 + norm (X M)" in exI)
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
done

subsubsection {* Cauchy Sequences are Convergent *}

axclass banach \<subseteq> real_normed_vector
Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"

theorem LIMSEQ_imp_Cauchy:
assumes X: "X ----> a" shows "Cauchy X"
proof (rule metric_CauchyI)
fix e::real assume "0 < e"
hence "0 < e/2" by simp
with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
proof (intro exI allI impI)
fix m assume "N \<le> m"
hence m: "dist (X m) a < e/2" using N by fast
fix n assume "N \<le> n"
hence n: "dist (X n) a < e/2" using N by fast
have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
by (rule dist_triangle2)
also from m n have "\<dots> < e" by simp
finally show "dist (X m) (X n) < e" .
qed
qed

lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
unfolding convergent_def
by (erule exE, erule LIMSEQ_imp_Cauchy)

text {*
Proof that Cauchy sequences converge based on the one from
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
*}

text {*
If sequence @{term "X"} is Cauchy, then its limit is the lub of
@{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
*}

lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"

lemma real_abs_diff_less_iff:
"(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
by auto

locale real_Cauchy =
fixes X :: "nat \<Rightarrow> real"
assumes X: "Cauchy X"
fixes S :: "real set"
defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"

lemma real_CauchyI:
assumes "Cauchy X"
shows "real_Cauchy X"
proof qed (fact assms)

lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
by (unfold S_def, auto)

lemma (in real_Cauchy) bound_isUb:
assumes N: "\<forall>n\<ge>N. X n < x"
shows "isUb UNIV S x"
proof (rule isUb_UNIV_I)
fix y::real assume "y \<in> S"
hence "\<exists>M. \<forall>n\<ge>M. y < X n"
then obtain M where "\<forall>n\<ge>M. y < X n" ..
hence "y < X (max M N)" by simp
also have "\<dots> < x" using N by simp
finally show "y \<le> x"
by (rule order_less_imp_le)
qed

lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
proof (rule reals_complete)
obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
using CauchyD [OF X zero_less_one] by fast
hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
show "\<exists>x. x \<in> S"
proof
from N have "\<forall>n\<ge>N. X N - 1 < X n"
thus "X N - 1 \<in> S" by (rule mem_S)
qed
show "\<exists>u. isUb UNIV S u"
proof
from N have "\<forall>n\<ge>N. X n < X N + 1"
thus "isUb UNIV S (X N + 1)"
by (rule bound_isUb)
qed
qed

lemma (in real_Cauchy) isLub_imp_LIMSEQ:
assumes x: "isLub UNIV S x"
shows "X ----> x"
proof (rule LIMSEQ_I)
fix r::real assume "0 < r"
hence r: "0 < r/2" by simp
obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
using CauchyD [OF X r] by fast
hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
by (simp only: real_norm_def real_abs_diff_less_iff)

from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
hence "X N - r/2 \<in> S" by (rule mem_S)
hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast

from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast

show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
proof (intro exI allI impI)
fix n assume n: "N \<le> n"
from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
thus "norm (X n - x) < r" using 1 2
qed
qed

lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
proof -
obtain x where "isLub UNIV S x"
using isLub_ex by fast
hence "X ----> x"
by (rule isLub_imp_LIMSEQ)
thus ?thesis ..
qed

lemma real_Cauchy_convergent:
fixes X :: "nat \<Rightarrow> real"
shows "Cauchy X \<Longrightarrow> convergent X"
unfolding convergent_def
by (rule real_Cauchy.LIMSEQ_ex)
(rule real_CauchyI)

instance real :: banach
by intro_classes (rule real_Cauchy_convergent)

lemma Cauchy_convergent_iff:
fixes X :: "nat \<Rightarrow> 'a::banach"
shows "Cauchy X = convergent X"
by (fast intro: Cauchy_convergent convergent_Cauchy)

lemma convergent_subseq_convergent:
fixes X :: "nat \<Rightarrow> 'a::banach"
shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])

subsection {* Power Sequences *}

text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
also fact that bounded and monotonic sequence converges.*}

lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
apply (rule_tac x = 1 in exI)
apply (auto dest: power_mono)
done

lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
apply (clarify intro!: mono_SucI2)
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
done

lemma convergent_realpow:
"[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)

lemma LIMSEQ_inverse_realpow_zero_lemma:
fixes x :: real
assumes x: "0 \<le> x"
shows "real n * x + 1 \<le> (x + 1) ^ n"
apply (induct n)
apply simp
apply simp
apply (rule order_trans)
prefer 2
apply (erule mult_left_mono)
apply (rule add_increasing [OF x], simp)
done

lemma LIMSEQ_inverse_realpow_zero:
"1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
proof (rule LIMSEQ_inverse_zero [rule_format])
fix y :: real
assume x: "1 < x"
hence "0 < x - 1" by simp
hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
by (rule reals_Archimedean3)
hence "\<exists>N::nat. y < real N * (x - 1)" ..
then obtain N::nat where "y < real N * (x - 1)" ..
also have "\<dots> \<le> real N * (x - 1) + 1" by simp
also have "\<dots> \<le> (x - 1 + 1) ^ N"
by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
also have "\<dots> = x ^ N" by simp
finally have "y < x ^ N" .
hence "\<forall>n\<ge>N. y < x ^ n"
apply clarify
apply (erule order_less_le_trans)
apply (erule power_increasing)
apply (rule order_less_imp_le [OF x])
done
thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
qed

lemma LIMSEQ_realpow_zero:
"\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
proof (cases)
assume "x = 0"
hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
thus ?thesis by (rule LIMSEQ_imp_Suc)
next
assume "0 \<le> x" and "x \<noteq> 0"
hence x0: "0 < x" by simp
assume x1: "x < 1"
from x0 x1 have "1 < inverse x"
by (rule real_inverse_gt_one)
hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
by (rule LIMSEQ_inverse_realpow_zero)
thus ?thesis by (simp add: power_inverse)
qed

lemma LIMSEQ_power_zero:
fixes x :: "'a::{real_normed_algebra_1}"
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
done

lemma LIMSEQ_divide_realpow_zero:
"1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
apply (cut_tac a = a and x1 = "inverse x" in
LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
apply (auto simp add: divide_inverse power_inverse)
done

text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}

lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])

lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
apply (rule LIMSEQ_rabs_zero [THEN iffD1])
apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
done

end
```