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convergence of functions wrt deleted nbhs #1385

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Nov 9, 2024
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convergence of functions wrt deleted nbhs #1385

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convergence of functions wrt deleted nbhs
affeldt-aist Nov 7, 2024
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affeldt-aist Nov 8, 2024
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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -69,6 +69,8 @@

- in `measure.v`:
+ lemma `countable_measurable`
- in `realfun.v`:
+ lemma `cvg_dnbhsP`

### Changed

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14 changes: 14 additions & 0 deletions theories/realfun.v
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Expand Up @@ -207,6 +207,20 @@ apply: pfl.
by split; [move=> k; rewrite ltrNl//|apply/cvgNP => /=; rewrite opprK].
Qed.

Lemma cvg_dnbhsP (f : R -> R) (p l : R) :
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f x @[x --> p^'] --> l <->
(forall u : R^nat, (forall n, u n != p) /\ (u n @[n --> \oo] --> p) ->
f (u n) @[n --> \oo] --> l).
Proof.
split=> [/cvgrPdist_le fpl u [up /cvgrPdist_lt put]|pfl].
apply/cvgrPdist_le => e /fpl [r /=] /put[n _ {}put] {}fpl.
near=> t; apply: fpl => //=; apply: put.
by near: t; exact: nbhs_infty_ge.
apply: cvg_at_right_left_dnbhs.
- by apply/cvg_at_rightP => u [pu ?]; apply: pfl; split => // n; rewrite gt_eqF.
- by apply/cvg_at_leftP => u [pu ?]; apply: pfl; split => // n; rewrite lt_eqF.
Unshelve. all: end_near. Qed.

End cvgr_fun_cvg_seq.

Section cvge_fun_cvg_seq.
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