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Mirrors > Home > MPE Home > Th. List > caurcvg | Structured version Visualization version GIF version |
Description: A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by AV, 12-Sep-2020.) |
Ref | Expression |
---|---|
caurcvg.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
caurcvg.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
caurcvg.4 | ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
Ref | Expression |
---|---|
caurcvg | ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caurcvg.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | uzssz 12874 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 4014 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
4 | zssre 12596 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
5 | 3, 4 | sstri 3989 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
7 | caurcvg.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
8 | 1rp 13011 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
9 | 8 | ne0ii 4338 | . . . . 5 ⊢ ℝ+ ≠ ∅ |
10 | caurcvg.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
11 | r19.2z 4495 | . . . . 5 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
12 | 9, 10, 11 | sylancr 586 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
13 | eluzel2 12858 | . . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
14 | 13, 1 | eleq2s 2847 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ) |
15 | 1 | uzsup 13861 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞) |
16 | 14, 15 | syl 17 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → sup(𝑍, ℝ*, < ) = +∞) |
17 | 16 | a1d 25 | . . . . . 6 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞)) |
18 | 17 | rexlimiv 3145 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞) |
19 | 18 | rexlimivw 3148 | . . . 4 ⊢ (∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞) |
20 | 12, 19 | syl 17 | . . 3 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞) |
21 | 3 | sseli 3976 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
22 | 3 | sseli 3976 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
23 | eluz 12867 | . . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) | |
24 | 21, 22, 23 | syl2an 595 | . . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) |
25 | 24 | biimprd 247 | . . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑚 ≤ 𝑘 → 𝑘 ∈ (ℤ≥‘𝑚))) |
26 | 25 | expimpd 453 | . . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ∧ 𝑚 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑚))) |
27 | 26 | imim1d 82 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑚) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) → ((𝑘 ∈ 𝑍 ∧ 𝑚 ≤ 𝑘) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))) |
28 | 27 | exp4a 431 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑚) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) → (𝑘 ∈ 𝑍 → (𝑚 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)))) |
29 | 28 | ralimdv2 3160 | . . . . . 6 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → ∀𝑘 ∈ 𝑍 (𝑚 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))) |
30 | 29 | reximia 3078 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑚 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
31 | 30 | ralimi 3080 | . . . 4 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑚 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
32 | 10, 31 | syl 17 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑚 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
33 | 6, 7, 20, 32 | caurcvgr 15653 | . 2 ⊢ (𝜑 → 𝐹 ⇝𝑟 (lim sup‘𝐹)) |
34 | 14 | a1d 25 | . . . . . 6 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ)) |
35 | 34 | rexlimiv 3145 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ) |
36 | 35 | rexlimivw 3148 | . . . 4 ⊢ (∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ) |
37 | 12, 36 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
38 | ax-resscn 11196 | . . . 4 ⊢ ℝ ⊆ ℂ | |
39 | fss 6739 | . . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑍⟶ℂ) | |
40 | 7, 38, 39 | sylancl 585 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
41 | 1, 37, 40 | rlimclim 15523 | . 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 (lim sup‘𝐹) ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
42 | 33, 41 | mpbid 231 | 1 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3947 ∅c0 4323 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supcsup 9464 ℂcc 11137 ℝcr 11138 1c1 11140 +∞cpnf 11276 ℝ*cxr 11278 < clt 11279 ≤ cle 11280 − cmin 11475 ℤcz 12589 ℤ≥cuz 12853 ℝ+crp 13007 abscabs 15214 lim supclsp 15447 ⇝ cli 15461 ⇝𝑟 crli 15462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-ico 13363 df-fl 13790 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 |
This theorem is referenced by: caurcvg2 15657 mbflimlem 25609 climlimsup 45148 ioodvbdlimc1lem1 45319 |
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