adherence value, extracted seq, limit point #1780
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affeldt-aist
added
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enhancement ✨
affeldt-aist
commented
Nov 28, 2025
| Let A_nonempty x : A (N_ x) (idx_ x) !=set0. | ||
| Proof. by move: x => [[? ?]] []. Qed. | ||
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| Let elt_rel i j := [/\ (idx_ j = (idx_ i).+1), |
affeldt-aist
commented
Nov 28, 2025
| Qed. | ||
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| Let adherence_value_cvg_direct : adherence_value u_ a -> | ||
| exists2 f : nat -> nat, increasing_seq f & (u_ \o f @ \oo --> a). |
Member
Author
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Note that Bolzano-Weierstrass is a direct consequence of these lemmas. |
Tragicus
reviewed
Dec 3, 2025
| Qed. | ||
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| Definition adherence_value {R : numDomainType} (u_ : R^nat) (a : R) := | ||
| forall (e : R) n, e > 0 -> exists2 p, (p >= n)%N & `|u_ p - a| <= e. |
Collaborator
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I am wondering if we would prefer a \in closure (range u_ : set R^o).
| Proof. | ||
| move=> A0. | ||
| pose B : set int := [set x%:~R | x in A]. | ||
| have B0 : B !=set0 by case: A0 => a Aa; exists a%:~R, a. |
Collaborator
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Suggested change
| have B0 : B !=set0 by case: A0 => a Aa; exists a%:~R, a. | |
| have B0 : B !=set0 by apply: image_nonempty. |
golfing
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| have : lbound B 0 by move=> _ [b0 Bb0 <-]; rewrite ler0z. | ||
| move/(int_lbound_has_minimum B0) => [_ [[i Ai <-]]] Bi. | ||
| exists i; split => // k Bk. |
Collaborator
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Suggested change
| have : lbound B 0 by move=> _ [b0 Bb0 <-]; rewrite ler0z. | |
| move/(int_lbound_has_minimum B0) => [_ [[i Ai <-]]] Bi. | |
| exists i; split => // k Bk. | |
| have: lbound B 0 by move=> _ [b0 Bb0 <-]; rewrite ler0z. | |
| move/(int_lbound_has_minimum B0) => [_ [[i Ai <-]]] Bi. | |
| exists i; split=> // k Bk. |
Collaborator
How so? |
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CHANGELOG_UNRELEASED.mdReference: How to document
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