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convergence of a real function #1396

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convergence of a real function #1396

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -35,6 +35,9 @@
- in `probability.v`
+ lemma `integral_distribution` (existsing lemma `integral_distribution` has been renamed)

- in `realfun.v`:
+ lemma `cvgr_nbhsP`

### Changed

- in `lebesgue_integrale.v`
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42 changes: 36 additions & 6 deletions theories/realfun.v
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Expand Up @@ -154,9 +154,9 @@ End fun_cvg_realFieldType.

Section cvgr_fun_cvg_seq.
Context {R : realType}.
Implicit Types (f : R -> R) (p l : R).

Lemma cvg_at_rightP (f : R -> R) (p l : R) :
f x @[x --> p^'+] --> l <->
Lemma cvg_at_rightP f p l : 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.
Expand Down Expand Up @@ -193,8 +193,7 @@ rewrite /sval/=; case: cid => // x [px xpn].
by rewrite leNgt distrC => /negP.
Unshelve. all: by end_near. Qed.

Lemma cvg_at_leftP (f : R -> R) (p l : R) :
f x @[x --> p^'-] --> l <->
Lemma cvg_at_leftP f p l : 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.
Expand All @@ -207,8 +206,7 @@ apply: pfl.
by split; [move=> k; rewrite ltrNl//|apply/cvgNP => /=; rewrite opprK].
Qed.

Lemma cvgr_dnbhsP (f : R -> R) (p l : R) :
f x @[x --> p^'] --> l <->
Lemma cvgr_dnbhsP f p l : 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.
Expand All @@ -221,6 +219,38 @@ apply: cvg_at_right_left_dnbhs.
- by apply/cvg_at_leftP => u [pu ?]; apply: pfl; split => // n; rewrite lt_eqF.
Unshelve. all: end_near. Qed.

Lemma cvgr_nbhsP f p l : f x @[x --> p] --> l <->
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It would be nice to have this for any metric space (or indeed any first-countable space). But it's fine to have the specialized version for now if it's useful. We can always go back and generalize later

(forall u : R^nat, (u n @[n --> \oo] --> p) -> f (u n) @[n --> \oo] --> l).
Proof.
split=> [/cvgrPdist_le /= fpl u /cvgrPdist_lt /= uyp|pfl].
apply/cvgrPdist_le => e /fpl[d d0 pdf].
by apply: filterS (uyp d d0) => t /pdf.
apply: contrapT => fpl; move: pfl; apply/existsNP.
suff: exists2 x : R ^nat,
x n @[n --> \oo] --> p & ~ f (x n) @[n --> \oo] --> l.
by move=> [x_] hp; exists x_; exact/not_implyP.
have [e He] : exists e : {posnum R}, forall d : {posnum R},
exists xn, `|xn - p| < d%:num /\ `|f xn - l| >= e%:num.
apply: contrapT; apply: contra_not fpl => /forallNP h.
apply/cvgrPdist_le => e e0; have /existsNP[d] := h (PosNum e0).
move/forallNP => {}h; near=> t.
have /not_andP[abs|/negP] := h t.
- exfalso; apply: abs.
by near: t; exists d%:num => //= z/=; rewrite distrC.
- by rewrite -ltNge distrC => /ltW.
have invn n : 0 < n.+1%:R^-1 :> R by rewrite invr_gt0.
exists (fun n => sval (cid (He (PosNum (invn n))))).
apply/cvgrPdist_lt => r r0; near=> t.
rewrite /sval/=; case: cid => x [xpt _].
rewrite distrC (lt_le_trans xpt)// -[leRHS]invrK lef_pV2 ?posrE ?invr_gt0//.
near: t; exists `|ceil r^-1|%N => // s /=.
rewrite -ltnS -(@ltr_nat R) => /ltW; apply: le_trans.
by rewrite natr_absz gtr0_norm -?ceil_gt0 ?invr_gt0 ?le_ceil ?num_real.
move=> /cvgrPdist_lt/(_ e%:num (ltac:(by [])))[] n _ /(_ _ (leqnn _)).
rewrite /sval/=; case: cid => // x [px xpn].
by rewrite ltNge distrC => /negP.
Unshelve. all: end_near. Qed.

End cvgr_fun_cvg_seq.

Section cvge_fun_cvg_seq.
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