![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12252 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 3949 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
4 | zssre 11976 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
5 | 3, 4 | sstri 3924 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
7 | caurcvg.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
8 | 1rp 12381 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
9 | 8 | ne0ii 4253 | . . . . 5 ⊢ ℝ+ ≠ ∅ |
10 | caurcvg.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
11 | r19.2z 4398 | . . . . 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 590 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
13 | eluzel2 12236 | . . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
14 | 13, 1 | eleq2s 2908 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ) |
15 | 1 | uzsup 13226 | . . . . . . . 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 3239 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞) |
19 | 18 | rexlimivw 3241 | . . . 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 3911 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
22 | 3 | sseli 3911 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
23 | eluz 12245 | . . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) | |
24 | 21, 22, 23 | syl2an 598 | . . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) |
25 | 24 | biimprd 251 | . . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑚 ≤ 𝑘 → 𝑘 ∈ (ℤ≥‘𝑚))) |
26 | 25 | expimpd 457 | . . . . . . . . 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 435 | . . . . . . 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 3143 | . . . . . 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 3205 | . . . . 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 3128 | . . . 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 15022 | . 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 3239 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ) |
36 | 35 | rexlimivw 3241 | . . . 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 10583 | . . . 4 ⊢ ℝ ⊆ ℂ | |
39 | fss 6501 | . . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑍⟶ℂ) | |
40 | 7, 38, 39 | sylancl 589 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
41 | 1, 37, 40 | rlimclim 14895 | . 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 (lim sup‘𝐹) ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
42 | 33, 41 | mpbid 235 | 1 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 1c1 10527 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 lim supclsp 14819 ⇝ cli 14833 ⇝𝑟 crli 14834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 |
This theorem is referenced by: caurcvg2 15026 mbflimlem 24271 climlimsup 42402 ioodvbdlimc1lem1 42573 |
Copyright terms: Public domain | W3C validator |