# Theory Uniform_Limit

```(*  Title:      HOL/Analysis/Uniform_Limit.thy
Author:     Christoph Traut, TU München
Author:     Fabian Immler, TU München
*)

section ‹Uniform Limit and Uniform Convergence›

theory Uniform_Limit
imports Connected Summation_Tests Infinite_Sum
begin

subsection ‹Definition›

definition✐‹tag important› uniformly_on :: "'a set ⇒ ('a ⇒ 'b::metric_space) ⇒ ('a ⇒ 'b) filter"
where "uniformly_on S l = (INF e∈{0 <..}. principal {f. ∀x∈S. dist (f x) (l x) < e})"

abbreviation✐‹tag important›
"uniform_limit S f l ≡ filterlim f (uniformly_on S l)"

definition uniformly_convergent_on where
"uniformly_convergent_on X f ⟷ (∃l. uniform_limit X f l sequentially)"

definition uniformly_Cauchy_on where
"uniformly_Cauchy_on X f ⟷ (∀e>0. ∃M. ∀x∈X. ∀(m::nat)≥M. ∀n≥M. dist (f m x) (f n x) < e)"

proposition uniform_limit_iff:
"uniform_limit S f l F ⟷ (∀e>0. ∀⇩F n in F. ∀x∈S. dist (f n x) (l x) < e)"
unfolding filterlim_iff uniformly_on_def
by (subst eventually_INF_base)
(fastforce
simp: eventually_principal uniformly_on_def
intro: bexI[where x="min a b" for a b]
elim: eventually_mono)+

lemma uniform_limitD:
"uniform_limit S f l F ⟹ e > 0 ⟹ ∀⇩F n in F. ∀x∈S. dist (f n x) (l x) < e"

lemma uniform_limitI:
"(⋀e. e > 0 ⟹ ∀⇩F n in F. ∀x∈S. dist (f n x) (l x) < e) ⟹ uniform_limit S f l F"

lemma uniform_limit_sequentially_iff:
"uniform_limit S f l sequentially ⟷ (∀e>0. ∃N. ∀n≥N. ∀x ∈ S. dist (f n x) (l x) < e)"
unfolding uniform_limit_iff eventually_sequentially ..

lemma uniform_limit_at_iff:
"uniform_limit S f l (at x) ⟷
(∀e>0. ∃d>0. ∀z. 0 < dist z x ∧ dist z x < d ⟶ (∀x∈S. dist (f z x) (l x) < e))"
unfolding uniform_limit_iff eventually_at by simp

lemma uniform_limit_at_le_iff:
"uniform_limit S f l (at x) ⟷
(∀e>0. ∃d>0. ∀z. 0 < dist z x ∧ dist z x < d ⟶ (∀x∈S. dist (f z x) (l x) ≤ e))"
unfolding uniform_limit_iff eventually_at
by (fastforce dest: spec[where x = "e / 2" for e])

lemma uniform_limit_compose:
assumes ul: "uniform_limit X g l F"
and cont: "uniformly_continuous_on U f"
and g: "∀⇩F n in F. g n ` X ⊆ U" and l: "l ` X ⊆ U"
shows "uniform_limit X (λa b. f (g a b)) (f ∘ l) F"
proof (rule uniform_limitI)
fix ε::real
assume "0 < ε"
then obtain δ where "δ > 0" and δ: "⋀u v. ⟦u∈U; v∈U; dist u v < δ⟧ ⟹ dist (f u) (f v) < ε"
by (metis cont uniformly_continuous_on_def)
show "∀⇩F n in F. ∀x∈X. dist (f (g n x)) ((f ∘ l) x) < ε"
using uniform_limitD [OF ul ‹δ>0›] g unfolding o_def
by eventually_elim (use δ l in blast)
qed

lemma metric_uniform_limit_imp_uniform_limit:
assumes f: "uniform_limit S f a F"
assumes le: "eventually (λx. ∀y∈S. dist (g x y) (b y) ≤ dist (f x y) (a y)) F"
shows "uniform_limit S g b F"
proof (rule uniform_limitI)
fix e :: real
assume "0 < e"
from uniform_limitD[OF f this] le
show "∀⇩F x in F. ∀y∈S. dist (g x y) (b y) < e"
by eventually_elim force
qed

subsection ‹Exchange limits›

proposition swap_uniform_limit:
assumes f: "∀⇩F n in F. (f n ⤏ g n) (at x within S)"
assumes g: "(g ⤏ l) F"
assumes uc: "uniform_limit S f h F"
assumes "¬trivial_limit F"
shows "(h ⤏ l) (at x within S)"
proof (rule tendstoI)
fix e :: real
define e' where "e' = e/3"
assume "0 < e"
then have "0 < e'" by (simp add: e'_def)
from uniform_limitD[OF uc ‹0 < e'›]
have "∀⇩F n in F. ∀x∈S. dist (h x) (f n x) < e'"
moreover
from f
have "∀⇩F n in F. ∀⇩F x in at x within S. dist (g n) (f n x) < e'"
by eventually_elim (auto dest!: tendstoD[OF _ ‹0 < e'›] simp: dist_commute)
moreover
from tendstoD[OF g ‹0 < e'›] have "∀⇩F x in F. dist l (g x) < e'"
ultimately
have "∀⇩F _ in F. ∀⇩F x in at x within S. dist (h x) l < e"
proof eventually_elim
case (elim n)
note fh = elim(1)
note gl = elim(3)
have "∀⇩F x in at x within S. x ∈ S"
by (auto simp: eventually_at_filter)
with elim(2)
show ?case
proof eventually_elim
case (elim x)
from fh[rule_format, OF ‹x ∈ S›] elim(1)
have "dist (h x) (g n) < e' + e'"
from dist_triangle_lt[OF add_strict_mono, OF this gl]
show ?case by (simp add: e'_def)
qed
qed
thus "∀⇩F x in at x within S. dist (h x) l < e"
using eventually_happens by (metis ‹¬trivial_limit F›)
qed

subsection ‹Uniform limit theorem›

lemma tendsto_uniform_limitI:
assumes "uniform_limit S f l F"
assumes "x ∈ S"
shows "((λy. f y x) ⤏ l x) F"
using assms
by (auto intro!: tendstoI simp: eventually_mono dest!: uniform_limitD)

theorem uniform_limit_theorem:
assumes c: "∀⇩F n in F. continuous_on A (f n)"
assumes ul: "uniform_limit A f l F"
assumes "¬ trivial_limit F"
shows "continuous_on A l"
unfolding continuous_on_def
proof safe
fix x assume "x ∈ A"
then have "∀⇩F n in F. (f n ⤏ f n x) (at x within A)" "((λn. f n x) ⤏ l x) F"
using c ul
by (auto simp: continuous_on_def eventually_mono tendsto_uniform_limitI)
then show "(l ⤏ l x) (at x within A)"
by (rule swap_uniform_limit) fact+
qed

lemma uniformly_Cauchy_onI:
assumes "⋀e. e > 0 ⟹ ∃M. ∀x∈X. ∀m≥M. ∀n≥M. dist (f m x) (f n x) < e"
shows   "uniformly_Cauchy_on X f"
using assms unfolding uniformly_Cauchy_on_def by blast

lemma uniformly_Cauchy_onI':
assumes "⋀e. e > 0 ⟹ ∃M. ∀x∈X. ∀m≥M. ∀n>m. dist (f m x) (f n x) < e"
shows   "uniformly_Cauchy_on X f"
proof (rule uniformly_Cauchy_onI)
fix e :: real assume e: "e > 0"
from assms[OF this] obtain M
where M: "⋀x m n. x ∈ X ⟹ m ≥ M ⟹ n > m ⟹ dist (f m x) (f n x) < e" by fast
{
fix x m n assume x: "x ∈ X" and m: "m ≥ M" and n: "n ≥ M"
with M[OF this(1,2), of n] M[OF this(1,3), of m] e have "dist (f m x) (f n x) < e"
by (cases m n rule: linorder_cases) (simp_all add: dist_commute)
}
thus "∃M. ∀x∈X. ∀m≥M. ∀n≥M. dist (f m x) (f n x) < e" by fast
qed

lemma uniformly_Cauchy_imp_Cauchy:
"uniformly_Cauchy_on X f ⟹ x ∈ X ⟹ Cauchy (λn. f n x)"
unfolding Cauchy_def uniformly_Cauchy_on_def by fast

lemma uniform_limit_cong:
fixes f g :: "'a ⇒ 'b ⇒ ('c :: metric_space)" and h i :: "'b ⇒ 'c"
assumes "eventually (λy. ∀x∈X. f y x = g y x) F"
assumes "⋀x. x ∈ X ⟹ h x = i x"
shows   "uniform_limit X f h F ⟷ uniform_limit X g i F"
proof -
{
fix f g :: "'a ⇒ 'b ⇒ 'c" and h i :: "'b ⇒ 'c"
assume C: "uniform_limit X f h F" and A: "eventually (λy. ∀x∈X. f y x = g y x) F"
and B: "⋀x. x ∈ X ⟹ h x = i x"
{
fix e ::real assume "e > 0"
with C have "eventually (λy. ∀x∈X. dist (f y x) (h x) < e) F"
unfolding uniform_limit_iff by blast
with A have "eventually (λy. ∀x∈X. dist (g y x) (i x) < e) F"
by eventually_elim (insert B, simp_all)
}
hence "uniform_limit X g i F" unfolding uniform_limit_iff by blast
} note A = this
show ?thesis by (rule iffI) (erule A; insert assms; simp add: eq_commute)+
qed

lemma uniform_limit_cong':
fixes f g :: "'a ⇒ 'b ⇒ ('c :: metric_space)" and h i :: "'b ⇒ 'c"
assumes "⋀y x. x ∈ X ⟹ f y x = g y x"
assumes "⋀x. x ∈ X ⟹ h x = i x"
shows   "uniform_limit X f h F ⟷ uniform_limit X g i F"
using assms by (intro uniform_limit_cong always_eventually) blast+

lemma uniformly_convergent_cong:
assumes "eventually (λx. ∀y∈A. f x y = g x y) sequentially" "A = B"
shows "uniformly_convergent_on A f ⟷ uniformly_convergent_on B g"
unfolding uniformly_convergent_on_def assms(2) [symmetric]
by (intro iff_exI uniform_limit_cong eventually_mono [OF assms(1)]) auto

lemma uniformly_convergent_on_compose:
assumes "uniformly_convergent_on A f"
assumes "filterlim g sequentially sequentially"
shows   "uniformly_convergent_on A (λn. f (g n))"
proof -
from assms(1) obtain f' where "uniform_limit A f f' sequentially"
by (auto simp: uniformly_convergent_on_def)
hence "uniform_limit A (λn. f (g n)) f' sequentially"
by (rule filterlim_compose) fact
thus ?thesis
by (auto simp: uniformly_convergent_on_def)
qed

lemma uniformly_convergent_uniform_limit_iff:
"uniformly_convergent_on X f ⟷ uniform_limit X f (λx. lim (λn. f n x)) sequentially"
proof
assume "uniformly_convergent_on X f"
then obtain l where l: "uniform_limit X f l sequentially"
unfolding uniformly_convergent_on_def by blast
from l have "uniform_limit X f (λx. lim (λn. f n x)) sequentially ⟷
uniform_limit X f l sequentially"
by (intro uniform_limit_cong' limI tendsto_uniform_limitI[of f X l]) simp_all
also note l
finally show "uniform_limit X f (λx. lim (λn. f n x)) sequentially" .
qed (auto simp: uniformly_convergent_on_def)

lemma uniformly_convergentI: "uniform_limit X f l sequentially ⟹ uniformly_convergent_on X f"
unfolding uniformly_convergent_on_def by blast

lemma uniformly_convergent_on_empty [iff]: "uniformly_convergent_on {} f"

lemma uniformly_convergent_on_const [simp,intro]:
"uniformly_convergent_on A (λ_. c)"
by (auto simp: uniformly_convergent_on_def uniform_limit_iff intro!: exI[of _ c])

text‹Cauchy-type criteria for uniform convergence.›

lemma Cauchy_uniformly_convergent:
fixes f :: "nat ⇒ 'a ⇒ 'b :: complete_space"
assumes "uniformly_Cauchy_on X f"
shows   "uniformly_convergent_on X f"
unfolding uniformly_convergent_uniform_limit_iff uniform_limit_iff
proof safe
let ?f = "λx. lim (λn. f n x)"
fix e :: real assume e: "e > 0"
hence "e/2 > 0" by simp
with assms obtain N where N: "⋀x m n. x ∈ X ⟹ m ≥ N ⟹ n ≥ N ⟹ dist (f m x) (f n x) < e/2"
unfolding uniformly_Cauchy_on_def by fast
show "eventually (λn. ∀x∈X. dist (f n x) (?f x) < e) sequentially"
using eventually_ge_at_top[of N]
proof eventually_elim
fix n assume n: "n ≥ N"
show "∀x∈X. dist (f n x) (?f x) < e"
proof
fix x assume x: "x ∈ X"
with assms have "(λn. f n x) ⇢ ?f x"
by (auto dest!: Cauchy_convergent uniformly_Cauchy_imp_Cauchy simp: convergent_LIMSEQ_iff)
with ‹e/2 > 0› have "eventually (λm. m ≥ N ∧ dist (f m x) (?f x) < e/2) sequentially"
by (intro tendstoD eventually_conj eventually_ge_at_top)
then obtain m where m: "m ≥ N" "dist (f m x) (?f x) < e/2"
unfolding eventually_at_top_linorder by blast
have "dist (f n x) (?f x) ≤ dist (f n x) (f m x) + dist (f m x) (?f x)"
by (rule dist_triangle)
also from x n have "... < e/2 + e/2" by (intro add_strict_mono N m)
finally show "dist (f n x) (?f x) < e" by simp
qed
qed
qed

lemma uniformly_convergent_Cauchy:
assumes "uniformly_convergent_on X f"
shows "uniformly_Cauchy_on X f"
proof (rule uniformly_Cauchy_onI)
fix e::real assume "e > 0"
then have "0 < e / 2" by simp
with assms[unfolded uniformly_convergent_on_def uniform_limit_sequentially_iff]
obtain l N where l:"x ∈ X ⟹ n ≥ N ⟹ dist (f n x) (l x) < e / 2" for n x
by metis
from l l have "x ∈ X ⟹ n ≥ N ⟹ m ≥ N ⟹ dist (f n x) (f m x) < e" for n m x
by (rule dist_triangle_half_l)
then show "∃M. ∀x∈X. ∀m≥M. ∀n≥M. dist (f m x) (f n x) < e" by blast
qed

lemma uniformly_convergent_eq_Cauchy:
"uniformly_convergent_on X f = uniformly_Cauchy_on X f" for f::"nat ⇒ 'b ⇒ 'a::complete_space"
using Cauchy_uniformly_convergent uniformly_convergent_Cauchy by blast

lemma uniformly_convergent_eq_cauchy:
fixes s::"nat ⇒ 'b ⇒ 'a::complete_space"
shows
"(∃l. ∀e>0. ∃N. ∀n x. N ≤ n ∧ P x ⟶ dist(s n x)(l x) < e) ⟷
(∀e>0. ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x  ⟶ dist (s m x) (s n x) < e)"
proof -
have *: "(∀n≥N. ∀x. Q x ⟶ R n x) ⟷ (∀n x. N ≤ n ∧ Q x ⟶ R n x)"
"(∀x. Q x ⟶ (∀m≥N. ∀n≥N. S n m x)) ⟷ (∀m n x. N ≤ m ∧ N ≤ n ∧ Q x ⟶ S n m x)"
for N::nat and Q::"'b ⇒ bool" and R S
by blast+
show ?thesis
using uniformly_convergent_eq_Cauchy[of "Collect P" s]
unfolding uniformly_convergent_on_def uniformly_Cauchy_on_def uniform_limit_sequentially_iff
qed

lemma uniformly_cauchy_imp_uniformly_convergent:
fixes s :: "nat ⇒ 'a ⇒ 'b::complete_space"
assumes "∀e>0.∃N. ∀m (n::nat) x. N ≤ m ∧ N ≤ n ∧ P x --> dist(s m x)(s n x) < e"
and "∀x. P x --> (∀e>0. ∃N. ∀n. N ≤ n ⟶ dist(s n x)(l x) < e)"
shows "∀e>0. ∃N. ∀n x. N ≤ n ∧ P x ⟶ dist(s n x)(l x) < e"
proof -
obtain l' where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x ⟶ dist (s n x) (l' x) < e"
using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto
moreover
{
fix x
assume "P x"
then have "l x = l' x"
using tendsto_unique[OF trivial_limit_sequentially, of "λn. s n x" "l x" "l' x"]
using l and assms(2) unfolding lim_sequentially by blast
}
ultimately show ?thesis by auto
qed

lemma uniformly_convergent_on_sum_E:
fixes ε::real and f :: "nat ⇒ 'a ⇒ 'b::{complete_space,real_normed_vector}"
assumes uconv: "uniformly_convergent_on K (λn z. ∑k<n. f k z)" and "ε>0"
obtains N where "⋀m n z. ⟦N ≤ m; m ≤ n; z∈K⟧ ⟹ norm(∑k=m..<n. f k z) < ε"
proof -
obtain N where N: "⋀m n z. ⟦N ≤ m; N ≤ n; z∈K⟧ ⟹ dist (∑k<m. f k z) (∑k<n. f k z) < ε"
using uconv ‹ε>0› unfolding uniformly_Cauchy_on_def uniformly_convergent_eq_Cauchy by meson
show thesis
proof
fix m n z
assume "N ≤ m" "m ≤ n" "z ∈ K"
moreover have "(∑k = m..<n. f k z) = (∑k<n. f k z) - (∑k<m. f k z)"
by (metis atLeast0LessThan le0 sum_diff_nat_ivl ‹m ≤ n›)
ultimately show "norm(∑k = m..<n. f k z) < ε"
using N  by (simp add: dist_norm)
qed
qed

lemma uniformly_convergent_on_sum_iff:
fixes f :: "nat ⇒ 'a ⇒ 'b::{complete_space,real_normed_vector}"
shows "uniformly_convergent_on K (λn z. ∑k<n. f k z)
⟷ (∀ε>0. ∃N. ∀m n z. N≤m ⟶ m≤n ⟶ z∈K ⟶ norm (∑k=m..<n. f k z) < ε)" (is "?lhs=?rhs")
proof
assume R: ?rhs
show ?lhs
unfolding uniformly_Cauchy_on_def uniformly_convergent_eq_Cauchy
proof (intro strip)
fix ε::real
assume "ε>0"
then obtain N where "⋀m n z. ⟦N ≤ m; m ≤ n; z∈K⟧ ⟹ norm(∑k = m..<n. f k z) < ε"
using R by blast
then have "∀x∈K. ∀m≥N. ∀n≥m. norm ((∑k<m. f k x) - (∑k<n. f k x)) < ε"
by (metis atLeast0LessThan le0 sum_diff_nat_ivl norm_minus_commute)
then have "∀x∈K. ∀m≥N. ∀n≥N. norm ((∑k<m. f k x) - (∑k<n. f k x)) < ε"
by (metis linorder_le_cases norm_minus_commute)
then show "∃M. ∀x∈K. ∀m≥M. ∀n≥M. dist (∑k<m. f k x) (∑k<n. f k x) < ε"
by (metis dist_norm)
qed
qed (metis uniformly_convergent_on_sum_E)

lemma uniform_limit_suminf:
fixes f:: "nat ⇒ 'a::{metric_space, comm_monoid_add} ⇒ 'a"
assumes "uniformly_convergent_on X (λn x. ∑k<n. f k x)"
shows "uniform_limit X (λn x. ∑k<n. f k x) (λx. ∑k. f k x) sequentially"
proof -
obtain S where S: "uniform_limit X (λn x. ∑k<n. f k x) S sequentially"
using assms uniformly_convergent_on_def by blast
then have "(∑k. f k x) = S x" if "x ∈ X" for x
using that sums_iff sums_def by (blast intro: tendsto_uniform_limitI [OF S])
with S show ?thesis
by (simp cong: uniform_limit_cong')
qed

text ‹TODO: remove explicit formulations
@{thm uniformly_convergent_eq_cauchy uniformly_cauchy_imp_uniformly_convergent}?!›

lemma uniformly_convergent_imp_convergent:
"uniformly_convergent_on X f ⟹ x ∈ X ⟹ convergent (λn. f n x)"
unfolding uniformly_convergent_on_def convergent_def
by (auto dest: tendsto_uniform_limitI)

subsection ‹Comparison Test›

lemma uniformly_summable_comparison_test:
fixes f :: "nat ⇒ 'a ⇒ 'b :: banach"
assumes "uniformly_convergent_on A (λN x. ∑n<N. g n x)"
assumes "⋀n x. x ∈ A ⟹ norm (f n x) ≤ g n x"
shows   "uniformly_convergent_on A (λN x. ∑n<N. f n x)"
proof -
have "uniformly_Cauchy_on A (λN x. ∑n<N. f n x)"
proof (rule uniformly_Cauchy_onI')
fix e :: real assume e: "e > 0"
obtain M where M: "⋀x m n. x ∈ A ⟹ m ≥ M ⟹ n ≥ M ⟹ dist (∑k<m. g k x) (∑k<n. g k x) < e"
using assms(1) e unfolding uniformly_convergent_eq_Cauchy uniformly_Cauchy_on_def by metis
show "∃M. ∀x∈A. ∀m≥M. ∀n>m. dist (∑k<m. f k x) (∑k<n. f k x) < e"
proof (rule exI[of _ M], safe)
fix x m n assume xmn: "x ∈ A" "m ≥ M" "m < n"
have nonneg: "g k x ≥ 0" for k
by (rule order.trans[OF _ assms(2)]) (use xmn in auto)
have "dist (∑k<m. f k x) (∑k<n. f k x) = norm (∑k∈{..<n}-{..<m}. f k x)"
using xmn by (subst sum_diff) (auto simp: dist_norm norm_minus_commute)
also have "{..<n}-{..<m} = {m..<n}"
by auto
also have "norm (∑k∈{m..<n}. f k x) ≤ (∑k∈{m..<n}. norm (f k x))"
using norm_sum by blast
also have "… ≤ (∑k∈{m..<n}. g k x)"
by (intro sum_mono assms xmn)
also have "… = ¦∑k∈{m..<n}. g k x¦"
by (subst abs_of_nonneg) (auto simp: nonneg intro!: sum_nonneg)
also have "{m..<n} = {..<n} - {..<m}"
by auto
also have "¦∑k∈…. g k x¦ = dist (∑k<m. g k x) (∑k<n. g k x)"
using xmn by (subst sum_diff) (auto simp: abs_minus_commute dist_norm)
also have "… < e"
by (rule M) (use xmn in auto)
finally show "dist (∑k<m. f k x) (∑k<n. f k x) < e" .
qed
qed
thus ?thesis
unfolding uniformly_convergent_eq_Cauchy .
qed

lemma uniform_limit_compose_uniformly_continuous_on:
fixes f :: "'a :: metric_space ⇒ 'b :: metric_space"
assumes lim: "uniform_limit A g g' F"
assumes cont: "uniformly_continuous_on B f"
assumes ev: "eventually (λx. ∀y∈A. g x y ∈ B) F" and "closed B"
shows   "uniform_limit A (λx y. f (g x y)) (λy. f (g' y)) F"
proof (cases "F = bot")
case [simp]: False
show ?thesis
unfolding uniform_limit_iff
proof safe
fix e :: real assume e: "e > 0"

have g'_in_B: "g' y ∈ B" if "y ∈ A" for y
proof (rule Lim_in_closed_set)
show "eventually (λx. g x y ∈ B) F"
using ev by eventually_elim (use that in auto)
show "((λx. g x y) ⤏ g' y) F"
using lim that by (metis tendsto_uniform_limitI)
qed (use ‹closed B› in auto)

obtain d where d: "d > 0" "⋀x y. x ∈ B ⟹ y ∈ B ⟹ dist x y < d ⟹ dist (f x) (f y) < e"
using e cont unfolding uniformly_continuous_on_def by metis
from lim have "eventually (λx. ∀y∈A. dist (g x y) (g' y) < d) F"
unfolding uniform_limit_iff using ‹d > 0›  by meson
thus "eventually (λx. ∀y∈A. dist (f (g x y)) (f (g' y)) < e) F"
using assms(3)
proof eventually_elim
case (elim x)
show "∀y∈A. dist (f (g x y)) (f (g' y)) < e"
proof safe
fix y assume y: "y ∈ A"
show "dist (f (g x y)) (f (g' y)) < e"
proof (rule d)
show "dist (g x y) (g' y) < d"
using elim y by blast
qed (use y elim g'_in_B in auto)
qed
qed
qed
qed (auto simp: filterlim_def)

lemma uniformly_convergent_on_compose_uniformly_continuous_on:
fixes f :: "'a :: metric_space ⇒ 'b :: metric_space"
assumes lim: "uniformly_convergent_on A g"
assumes cont: "uniformly_continuous_on B f"
assumes ev: "eventually (λx. ∀y∈A. g x y ∈ B) sequentially" and "closed B"
shows   "uniformly_convergent_on A (λx y. f (g x y))"
proof -
from lim obtain g' where g': "uniform_limit A g g' sequentially"
by (auto simp: uniformly_convergent_on_def)
thus ?thesis
using uniform_limit_compose_uniformly_continuous_on[OF g' cont ev ‹closed B›]
by (auto simp: uniformly_convergent_on_def)
qed

subsection ‹Weierstrass M-Test›

proposition Weierstrass_m_test_ev:
fixes f :: "_ ⇒ _ ⇒ _ :: banach"
assumes "eventually (λn. ∀x∈A. norm (f n x) ≤ M n) sequentially"
assumes "summable M"
shows "uniform_limit A (λn x. ∑i<n. f i x) (λx. suminf (λi. f i x)) sequentially"
proof (rule uniform_limitI)
fix e :: real
assume "0 < e"
from suminf_exist_split[OF ‹0 < e› ‹summable M›]
have "∀⇩F k in sequentially. norm (∑i. M (i + k)) < e"
by (auto simp: eventually_sequentially)
with eventually_all_ge_at_top[OF assms(1)]
show "∀⇩F n in sequentially. ∀x∈A. dist (∑i<n. f i x) (∑i. f i x) < e"
proof eventually_elim
case (elim k)
show ?case
proof safe
fix x assume "x ∈ A"
have "∃N. ∀n≥N. norm (f n x) ≤ M n"
using assms(1) ‹x ∈ A› by (force simp: eventually_at_top_linorder)
hence summable_norm_f: "summable (λn. norm (f n x))"
by(rule summable_norm_comparison_test[OF _ ‹summable M›])
have summable_f: "summable (λn. f n x)"
using summable_norm_cancel[OF summable_norm_f] .
have summable_norm_f_plus_k: "summable (λi. norm (f (i + k) x))"
using summable_ignore_initial_segment[OF summable_norm_f]
by auto
have summable_M_plus_k: "summable (λi. M (i + k))"
using summable_ignore_initial_segment[OF ‹summable M›]
by auto

have "dist (∑i<k. f i x) (∑i. f i x) = norm ((∑i. f i x) - (∑i<k. f i x))"
using dist_norm dist_commute by (subst dist_commute)
also have "... = norm (∑i. f (i + k) x)"
using suminf_minus_initial_segment[OF summable_f, where k=k] by simp
also have "... ≤ (∑i. norm (f (i + k) x))"
using summable_norm[OF summable_norm_f_plus_k] .
also have "... ≤ (∑i. M (i + k))"
by (rule suminf_le[OF _ summable_norm_f_plus_k summable_M_plus_k])
(insert elim(1) ‹x ∈ A›, simp)
finally show "dist (∑i<k. f i x) (∑i. f i x) < e"
using elim by auto
qed
qed
qed

text‹Alternative version, formulated as in HOL Light›
corollary✐‹tag unimportant› series_comparison_uniform:
fixes f :: "_ ⇒ nat ⇒ _ :: banach"
assumes g: "summable g" and le: "⋀n x. N ≤ n ∧ x ∈ A ⟹ norm(f x n) ≤ g n"
shows "∃l. ∀e. 0 < e ⟶ (∃N. ∀n x. N ≤ n ∧ x ∈ A ⟶ dist(sum (f x) {..<n}) (l x) < e)"
proof -
have 1: "∀⇩F n in sequentially. ∀x∈A. norm (f x n) ≤ g n"
using le eventually_sequentially by auto
show ?thesis
apply (rule_tac x="(λx. ∑i. f x i)" in exI)
apply (metis (no_types, lifting) eventually_sequentially uniform_limitD [OF Weierstrass_m_test_ev [OF 1 g]])
done
qed

corollary✐‹tag unimportant› Weierstrass_m_test:
fixes f :: "_ ⇒ _ ⇒ _ :: banach"
assumes "⋀n x. x ∈ A ⟹ norm (f n x) ≤ M n"
assumes "summable M"
shows "uniform_limit A (λn x. ∑i<n. f i x) (λx. suminf (λi. f i x)) sequentially"
using assms by (intro Weierstrass_m_test_ev always_eventually) auto

corollary✐‹tag unimportant› Weierstrass_m_test'_ev:
fixes f :: "_ ⇒ _ ⇒ _ :: banach"
assumes "eventually (λn. ∀x∈A. norm (f n x) ≤ M n) sequentially" "summable M"
shows   "uniformly_convergent_on A (λn x. ∑i<n. f i x)"
unfolding uniformly_convergent_on_def by (rule exI, rule Weierstrass_m_test_ev[OF assms])

corollary✐‹tag unimportant› Weierstrass_m_test':
fixes f :: "_ ⇒ _ ⇒ _ :: banach"
assumes "⋀n x. x ∈ A ⟹ norm (f n x) ≤ M n" "summable M"
shows   "uniformly_convergent_on A (λn x. ∑i<n. f i x)"
unfolding uniformly_convergent_on_def by (rule exI, rule Weierstrass_m_test[OF assms])

lemma Weierstrass_m_test_general:
fixes f :: "'a ⇒ 'b ⇒ 'c :: banach"
fixes M :: "'a ⇒ real"
assumes norm_le:  "⋀x y. x ∈ X ⟹ y ∈ Y ⟹ norm (f x y) ≤ M x"
assumes summable: "M summable_on X"
shows "uniform_limit Y (λX y. ∑x∈X. f x y) (λy. ∑⇩∞x∈X. f x y) (finite_subsets_at_top X)"
proof (rule uniform_limitI)
fix ε :: real
assume "ε > 0"
define S where "S = (λy. ∑⇩∞x∈X. f x y)"
have S: "((λx. f x y) has_sum S y) X" if y: "y ∈ Y" for y
unfolding S_def
proof (rule has_sum_infsum)
have "(λx. norm (f x y)) summable_on X"
by (rule abs_summable_on_comparison_test'[OF summable norm_le]) (use y in auto)
thus "(λx. f x y) summable_on X"
by (metis abs_summable_summable)
qed

define T where "T = (∑⇩∞x∈X. M x)"
have T: "(M has_sum T) X"
unfolding T_def by (simp add: local.summable)
have M_summable: "M summable_on X'" if "X' ⊆ X" for X'
using local.summable summable_on_subset_banach that by blast

have f_summable: "(λx. f x y) summable_on X'" if "X' ⊆ X" "y ∈ Y" for X' y
using S summable_on_def summable_on_subset_banach that by blast
have "eventually (λX'. dist (∑x∈X'. M x) T < ε) (finite_subsets_at_top X)"
using T ‹ε > 0› unfolding T_def has_sum_def tendsto_iff by blast
moreover have "eventually (λX'. finite X' ∧ X' ⊆ X) (finite_subsets_at_top X)"
ultimately show "∀⇩F X' in finite_subsets_at_top X. ∀y∈Y. dist (∑x∈X'. f x y) (∑⇩∞x∈X. f x y) < ε"
proof eventually_elim
case X': (elim X')
show "∀y∈Y. dist (∑x∈X'. f x y) (∑⇩∞x∈X. f x y) < ε"
proof
fix y assume y: "y ∈ Y"
have 1: "((λx. f x y) has_sum (S y - (∑x∈X'. f x y))) (X - X')"
using X' y by (intro has_sum_Diff S has_sum_finite[of X'] f_summable) auto
have 2: "(M has_sum (T - (∑x∈X'. M x))) (X - X')"
using X' y by (intro has_sum_Diff T has_sum_finite[of X'] M_summable) auto
have "dist (∑x∈X'. f x y) (∑⇩∞x∈X. f x y) = norm (S y - (∑x∈X'. f x y))"
by (simp add: dist_norm norm_minus_commute S_def)
also have "norm (S y - (∑x∈X'. f x y)) ≤ (∑⇩∞x∈X-X'. M x)"
using 2 y by (intro norm_infsum_le[OF 1 _ norm_le]) (auto simp: infsumI)
also have "… = T - (∑x∈X'. M x)"
using 2 by (auto simp: infsumI)
also have "… < ε"
using X' by (simp add: dist_norm)
finally show "dist (∑x∈X'. f x y) (∑⇩∞x∈X. f x y) < ε" .
qed
qed
qed

subsection✐‹tag unimportant› ‹Structural introduction rules›

lemma uniform_limit_eq_rhs: "uniform_limit X f l F ⟹ l = m ⟹ uniform_limit X f m F"
by simp

named_theorems uniform_limit_intros "introduction rules for uniform_limit"
setup ‹
fn context =>
Named_Theorems.get (Context.proof_of context) \<^named_theorems>‹uniform_limit_intros›
|> map_filter (try (fn thm => @{thm uniform_limit_eq_rhs} OF [thm])))
›

lemma (in bounded_linear) uniform_limit[uniform_limit_intros]:
assumes "uniform_limit X g l F"
shows "uniform_limit X (λa b. f (g a b)) (λa. f (l a)) F"
proof (rule uniform_limitI)
fix e::real
from pos_bounded obtain K
where K: "⋀x y. dist (f x) (f y) ≤ K * dist x y" "K > 0"
by (auto simp: ac_simps dist_norm diff[symmetric])
assume "0 < e" with ‹K > 0› have "e / K > 0" by simp
from uniform_limitD[OF assms this]
show "∀⇩F n in F. ∀x∈X. dist (f (g n x)) (f (l x)) < e"
by eventually_elim (metis le_less_trans mult.commute pos_less_divide_eq K)
qed

lemma (in bounded_linear) uniformly_convergent_on:
assumes "uniformly_convergent_on A g"
shows   "uniformly_convergent_on A (λx y. f (g x y))"
proof -
from assms obtain l where "uniform_limit A g l sequentially"
unfolding uniformly_convergent_on_def by blast
hence "uniform_limit A (λx y. f (g x y)) (λx. f (l x)) sequentially"
by (rule uniform_limit)
thus ?thesis unfolding uniformly_convergent_on_def by blast
qed

lemmas bounded_linear_uniform_limit_intros[uniform_limit_intros] =
bounded_linear.uniform_limit[OF bounded_linear_Im]
bounded_linear.uniform_limit[OF bounded_linear_Re]
bounded_linear.uniform_limit[OF bounded_linear_cnj]
bounded_linear.uniform_limit[OF bounded_linear_fst]
bounded_linear.uniform_limit[OF bounded_linear_snd]
bounded_linear.uniform_limit[OF bounded_linear_zero]
bounded_linear.uniform_limit[OF bounded_linear_of_real]
bounded_linear.uniform_limit[OF bounded_linear_inner_left]
bounded_linear.uniform_limit[OF bounded_linear_inner_right]
bounded_linear.uniform_limit[OF bounded_linear_divide]
bounded_linear.uniform_limit[OF bounded_linear_scaleR_right]
bounded_linear.uniform_limit[OF bounded_linear_mult_left]
bounded_linear.uniform_limit[OF bounded_linear_mult_right]
bounded_linear.uniform_limit[OF bounded_linear_scaleR_left]

lemmas uniform_limit_uminus[uniform_limit_intros] =
bounded_linear.uniform_limit[OF bounded_linear_minus[OF bounded_linear_ident]]

lemma uniform_limit_const[uniform_limit_intros]: "uniform_limit S (λx. c) c f"
by (auto intro!: uniform_limitI)

fixes f g::"'a ⇒ 'b ⇒ 'c::real_normed_vector"
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
shows "uniform_limit X (λa b. f a b + g a b) (λa. l a + m a) F"
proof (rule uniform_limitI)
fix e::real
assume "0 < e"
hence "0 < e / 2" by simp
from
uniform_limitD[OF assms(1) this]
uniform_limitD[OF assms(2) this]
show "∀⇩F n in F. ∀x∈X. dist (f n x + g n x) (l x + m x) < e"
qed

lemma uniform_limit_minus[uniform_limit_intros]:
fixes f g::"'a ⇒ 'b ⇒ 'c::real_normed_vector"
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
shows "uniform_limit X (λa b. f a b - g a b) (λa. l a - m a) F"
by (rule uniform_limit_intros assms)+

lemma uniform_limit_norm[uniform_limit_intros]:
assumes "uniform_limit S g l f"
shows "uniform_limit S (λx y. norm (g x y)) (λx. norm (l x)) f"
using assms
apply (rule metric_uniform_limit_imp_uniform_limit)
apply (rule eventuallyI)
by (metis dist_norm norm_triangle_ineq3 real_norm_def)

lemma (in bounded_bilinear) bounded_uniform_limit[uniform_limit_intros]:
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
assumes "bounded (m ` X)"
assumes "bounded (l ` X)"
shows "uniform_limit X (λa b. prod (f a b) (g a b)) (λa. prod (l a) (m a)) F"
proof (rule uniform_limitI)
fix e::real
from pos_bounded obtain K where K:
"0 < K" "⋀a b. norm (prod a b) ≤ norm a * norm b * K"
by auto
hence "sqrt (K*4) > 0" by simp

from assms obtain Km Kl
where Km: "Km > 0" "⋀x. x ∈ X ⟹ norm (m x) ≤ Km"
and Kl: "Kl > 0" "⋀x. x ∈ X ⟹ norm (l x) ≤ Kl"
by (auto simp: bounded_pos)
hence "K * Km * 4 > 0" "K * Kl * 4 > 0"
using ‹K > 0›
by simp_all
assume "0 < e"

hence "sqrt e > 0" by simp
from uniform_limitD[OF assms(1) divide_pos_pos[OF this ‹sqrt (K*4) > 0›]]
uniform_limitD[OF assms(2) divide_pos_pos[OF this ‹sqrt (K*4) > 0›]]
uniform_limitD[OF assms(1) divide_pos_pos[OF ‹e > 0› ‹K * Km * 4 > 0›]]
uniform_limitD[OF assms(2) divide_pos_pos[OF ‹e > 0› ‹K * Kl * 4 > 0›]]
show "∀⇩F n in F. ∀x∈X. dist (prod (f n x) (g n x)) (prod (l x) (m x)) < e"
proof eventually_elim
case (elim n)
show ?case
proof safe
fix x assume "x ∈ X"
have "dist (prod (f n x) (g n x)) (prod (l x) (m x)) ≤
norm (prod (f n x - l x) (g n x - m x)) +
norm (prod (f n x - l x) (m x)) +
norm (prod (l x) (g n x - m x))"
by (auto simp: dist_norm prod_diff_prod intro: order_trans norm_triangle_ineq add_mono)
also note K(2)[of "f n x - l x" "g n x - m x"]
also from elim(1)[THEN bspec, OF ‹_ ∈ X›, unfolded dist_norm]
have "norm (f n x - l x) ≤ sqrt e / sqrt (K * 4)"
by simp
also from elim(2)[THEN bspec, OF ‹_ ∈ X›, unfolded dist_norm]
have "norm (g n x - m x) ≤ sqrt e / sqrt (K * 4)"
by simp
also have "sqrt e / sqrt (K * 4) * (sqrt e / sqrt (K * 4)) * K = e / 4"
using ‹K > 0› ‹e > 0› by auto
also note K(2)[of "f n x - l x" "m x"]
also note K(2)[of "l x" "g n x - m x"]
also from elim(3)[THEN bspec, OF ‹_ ∈ X›, unfolded dist_norm]
have "norm (f n x - l x) ≤ e / (K * Km * 4)"
by simp
also from elim(4)[THEN bspec, OF ‹_ ∈ X›, unfolded dist_norm]
have "norm (g n x - m x) ≤ e / (K * Kl * 4)"
by simp
also note Kl(2)[OF ‹_ ∈ X›]
also note Km(2)[OF ‹_ ∈ X›]
also have "e / (K * Km * 4) * Km * K = e / 4"
using ‹K > 0› ‹Km > 0› by simp
also have " Kl * (e / (K * Kl * 4)) * K = e / 4"
using ‹K > 0› ‹Kl > 0› by simp
also have "e / 4 + e / 4 + e / 4 < e" using ‹e > 0› by simp
finally show "dist (prod (f n x) (g n x)) (prod (l x) (m x)) < e"
using ‹K > 0› ‹Kl > 0› ‹Km > 0› ‹e > 0›
by (simp add: algebra_simps mult_right_mono divide_right_mono)
qed
qed
qed

lemmas bounded_bilinear_bounded_uniform_limit_intros[uniform_limit_intros] =
bounded_bilinear.bounded_uniform_limit[OF Inner_Product.bounded_bilinear_inner]
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_mult]
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_scaleR]

lemma uniform_lim_mult:
fixes f :: "'a ⇒ 'b ⇒ 'c::real_normed_algebra"
assumes f: "uniform_limit S f l F"
and g: "uniform_limit S g m F"
and l: "bounded (l ` S)"
and m: "bounded (m ` S)"
shows "uniform_limit S (λa b. f a b * g a b) (λa. l a * m a) F"
by (intro bounded_bilinear_bounded_uniform_limit_intros assms)

lemma uniform_lim_inverse:
fixes f :: "'a ⇒ 'b ⇒ 'c::real_normed_field"
assumes f: "uniform_limit S f l F"
and b: "⋀x. x ∈ S ⟹ b ≤ norm(l x)"
and "b > 0"
shows "uniform_limit S (λx y. inverse (f x y)) (inverse ∘ l) F"
proof (rule uniform_limitI)
fix e::real
assume "e > 0"
have lte: "dist (inverse (f x y)) ((inverse ∘ l) y) < e"
if "b/2 ≤ norm (f x y)" "norm (f x y - l y) < e * b⇧2 / 2" "y ∈ S"
for x y
proof -
have [simp]: "l y ≠ 0" "f x y ≠ 0"
using ‹b > 0› that b [OF ‹y ∈ S›] by fastforce+
have "norm (l y - f x y) <  e * b⇧2 / 2"
by (metis norm_minus_commute that(2))
also have "... ≤ e * (norm (f x y) * norm (l y))"
using ‹e > 0› that b [OF ‹y ∈ S›] apply (simp add: power2_eq_square)
by (metis ‹b > 0› less_eq_real_def mult.left_commute mult_mono')
finally show ?thesis
by (auto simp: dist_norm field_split_simps norm_mult norm_divide)
qed
have "∀⇩F n in F. ∀x∈S. dist (f n x) (l x) < b/2"
using uniform_limitD [OF f, of "b/2"] by (simp add: ‹b > 0›)
then have "∀⇩F x in F. ∀y∈S. b/2 ≤ norm (f x y)"
apply (rule eventually_mono)
using b apply (simp only: dist_norm)
by (metis (no_types, opaque_lifting) diff_zero dist_commute dist_norm norm_triangle_half_l not_less)
then have "∀⇩F x in F. ∀y∈S. b/2 ≤ norm (f x y) ∧ norm (f x y - l y) < e * b⇧2 / 2"
apply (simp only: ball_conj_distrib dist_norm [symmetric])
apply (rule eventually_conj, assumption)
apply (rule uniform_limitD [OF f, of "e * b ^ 2 / 2"])
using ‹b > 0› ‹e > 0› by auto
then show "∀⇩F x in F. ∀y∈S. dist (inverse (f x y)) ((inverse ∘ l) y) < e"
using lte by (force intro: eventually_mono)
qed

lemma uniform_lim_divide:
fixes f :: "'a ⇒ 'b ⇒ 'c::real_normed_field"
assumes f: "uniform_limit S f l F"
and g: "uniform_limit S g m F"
and l: "bounded (l ` S)"
and b: "⋀x. x ∈ S ⟹ b ≤ norm(m x)"
and "b > 0"
shows "uniform_limit S (λa b. f a b / g a b) (λa. l a / m a) F"
proof -
have m: "bounded ((inverse ∘ m) ` S)"
using b ‹b > 0›
by (metis le_imp_inverse_le norm_inverse)
have "uniform_limit S (λa b. f a b * inverse (g a b))
(λa. l a * (inverse ∘ m) a) F"
by (rule uniform_lim_mult [OF f uniform_lim_inverse [OF g b ‹b > 0›] l m])
then show ?thesis
qed

lemma uniform_limit_null_comparison:
assumes "∀⇩F x in F. ∀a∈S. norm (f x a) ≤ g x a"
assumes "uniform_limit S g (λ_. 0) F"
shows "uniform_limit S f (λ_. 0) F"
using assms(2)
proof (rule metric_uniform_limit_imp_uniform_limit)
show "∀⇩F x in F. ∀y∈S. dist (f x y) 0 ≤ dist (g x y) 0"
using assms(1) by (rule eventually_mono) (force simp add: dist_norm)
qed

lemma uniform_limit_on_Un:
"uniform_limit I f g F ⟹ uniform_limit J f g F ⟹ uniform_limit (I ∪ J) f g F"
by (auto intro!: uniform_limitI dest!: uniform_limitD elim: eventually_elim2)

lemma uniform_limit_on_empty [iff]:
"uniform_limit {} f g F"
by (auto intro!: uniform_limitI)

lemma uniform_limit_on_UNION:
assumes "finite S"
assumes "⋀s. s ∈ S ⟹ uniform_limit (h s) f g F"
shows "uniform_limit (⋃(h ` S)) f g F"
using assms
by induct (auto intro: uniform_limit_on_empty uniform_limit_on_Un)

lemma uniform_limit_on_Union:
assumes "finite I"
assumes "⋀J. J ∈ I ⟹ uniform_limit J f g F"
shows "uniform_limit (Union I) f g F"
by (metis SUP_identity_eq assms uniform_limit_on_UNION)

lemma uniform_limit_on_subset:
"uniform_limit J f g F ⟹ I ⊆ J ⟹ uniform_limit I f g F"
by (auto intro!: uniform_limitI dest!: uniform_limitD intro: eventually_mono)

lemma uniform_limit_bounded:
fixes f::"'i ⇒ 'a::topological_space ⇒ 'b::metric_space"
assumes l: "uniform_limit S f l F"
assumes bnd: "∀⇩F i in F. bounded (f i ` S)"
assumes "F ≠ bot"
shows "bounded (l ` S)"
proof -
from l have "∀⇩F n in F. ∀x∈S. dist (l x) (f n x) < 1"
by (auto simp: uniform_limit_iff dist_commute dest!: spec[where x=1])
with bnd
have "∀⇩F n in F. ∃M. ∀x∈S. dist undefined (l x) ≤ M + 1"
by eventually_elim
(auto intro!: order_trans[OF dist_triangle2 add_mono] intro: less_imp_le
simp: bounded_any_center[where a=undefined])
then show ?thesis using assms
by (auto simp: bounded_any_center[where a=undefined] dest!: eventually_happens)
qed

"uniformly_convergent_on A f ⟹ uniformly_convergent_on A g⟹
uniformly_convergent_on A (λk x. f k x + g k x :: 'a :: {real_normed_algebra})"
unfolding uniformly_convergent_on_def by (blast dest: uniform_limit_add)

lemma uniformly_convergent_minus:
"uniformly_convergent_on A f ⟹ uniformly_convergent_on A g⟹
uniformly_convergent_on A (λk x. f k x - g k x :: 'a :: {real_normed_algebra})"
unfolding uniformly_convergent_on_def by (blast dest: uniform_limit_minus)

lemma uniformly_convergent_mult:
"uniformly_convergent_on A f ⟹
uniformly_convergent_on A (λk x. c * f k x :: 'a :: {real_normed_algebra})"
unfolding uniformly_convergent_on_def
by (blast dest: bounded_linear_uniform_limit_intros(13))

subsection‹Power series and uniform convergence›

proposition powser_uniformly_convergent:
fixes a :: "nat ⇒ 'a::{real_normed_div_algebra,banach}"
shows "uniformly_convergent_on (cball ξ r) (λn x. ∑i<n. a i * (x - ξ) ^ i)"
proof (cases "0 ≤ r")
case True
then have *: "summable (λn. norm (a n) * r ^ n)"
using abs_summable_in_conv_radius [of "of_real r" a] assms
show ?thesis
by (simp add: Weierstrass_m_test'_ev [OF _ *] norm_mult norm_power
mult_left_mono power_mono dist_norm norm_minus_commute)
next
case False then show ?thesis by (simp add: not_le)
qed

lemma powser_uniform_limit:
fixes a :: "nat ⇒ 'a::{real_normed_div_algebra,banach}"
shows "uniform_limit (cball ξ r) (λn x. ∑i<n. a i * (x - ξ) ^ i) (λx. suminf (λi. a i * (x - ξ) ^ i)) sequentially"
using powser_uniformly_convergent [OF assms]

lemma powser_continuous_suminf:
fixes a :: "nat ⇒ 'a::{real_normed_div_algebra,banach}"
shows "continuous_on (cball ξ r) (λx. suminf (λi. a i * (x - ξ) ^ i))"
apply (rule uniform_limit_theorem [OF _ powser_uniform_limit])
apply (rule eventuallyI continuous_intros assms)+
done

lemma powser_continuous_sums:
fixes a :: "nat ⇒ 'a::{real_normed_div_algebra,banach}"
assumes r: "r < conv_radius a"
and sm: "⋀x. x ∈ cball ξ r ⟹ (λn. a n * (x - ξ) ^ n) sums (f x)"
shows "continuous_on (cball ξ r) f"
apply (rule continuous_on_cong [THEN iffD1, OF refl _ powser_continuous_suminf [OF r]])
using sm sums_unique by fastforce

lemmas uniform_limit_subset_union = uniform_limit_on_subset[OF uniform_limit_on_Union]

end

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