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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 12896 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 4029 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
4 | zssre 12617 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
5 | 3, 4 | sstri 4004 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
7 | caurcvg.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
8 | 1rp 13035 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
9 | 8 | ne0ii 4349 | . . . . 5 ⊢ ℝ+ ≠ ∅ |
10 | caurcvg.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
11 | r19.2z 4500 | . . . . 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 587 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
13 | eluzel2 12880 | . . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
14 | 13, 1 | eleq2s 2856 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ) |
15 | 1 | uzsup 13899 | . . . . . . . 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 3990 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
22 | 3 | sseli 3990 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
23 | eluz 12889 | . . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) | |
24 | 21, 22, 23 | syl2an 596 | . . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) |
25 | 24 | biimprd 248 | . . . . . . . . . 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 15706 | . 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 11209 | . . . 4 ⊢ ℝ ⊆ ℂ | |
39 | fss 6752 | . . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑍⟶ℂ) | |
40 | 7, 38, 39 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
41 | 1, 37, 40 | rlimclim 15578 | . 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 (lim sup‘𝐹) ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
42 | 33, 41 | mpbid 232 | 1 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℂcc 11150 ℝcr 11151 1c1 11153 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 ℤcz 12610 ℤ≥cuz 12875 ℝ+crp 13031 abscabs 15269 lim supclsp 15502 ⇝ cli 15516 ⇝𝑟 crli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-fl 13828 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 |
This theorem is referenced by: caurcvg2 15710 mbflimlem 25715 climlimsup 45715 ioodvbdlimc1lem1 45886 |
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