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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 12267 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 4003 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
4 | zssre 11991 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
5 | 3, 4 | sstri 3978 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
7 | caurcvg.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
8 | 1rp 12396 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
9 | 8 | ne0ii 4305 | . . . . 5 ⊢ ℝ+ ≠ ∅ |
10 | caurcvg.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
11 | r19.2z 4442 | . . . . 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 589 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
13 | eluzel2 12251 | . . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
14 | 13, 1 | eleq2s 2933 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ) |
15 | 1 | uzsup 13234 | . . . . . . . 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 3282 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞) |
19 | 18 | rexlimivw 3284 | . . . 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 3965 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
22 | 3 | sseli 3965 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
23 | eluz 12260 | . . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) | |
24 | 21, 22, 23 | syl2an 597 | . . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) |
25 | 24 | biimprd 250 | . . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑚 ≤ 𝑘 → 𝑘 ∈ (ℤ≥‘𝑚))) |
26 | 25 | expimpd 456 | . . . . . . . . 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 434 | . . . . . . 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 3178 | . . . . . 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 3244 | . . . . 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 3162 | . . . 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 15032 | . 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 3282 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ) |
36 | 35 | rexlimivw 3284 | . . . 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 10596 | . . . 4 ⊢ ℝ ⊆ ℂ | |
39 | fss 6529 | . . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑍⟶ℂ) | |
40 | 7, 38, 39 | sylancl 588 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
41 | 1, 37, 40 | rlimclim 14905 | . 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 (lim sup‘𝐹) ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
42 | 33, 41 | mpbid 234 | 1 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℂcc 10537 ℝcr 10538 1c1 10540 +∞cpnf 10674 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 abscabs 14595 lim supclsp 14829 ⇝ cli 14843 ⇝𝑟 crli 14844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 |
This theorem is referenced by: caurcvg2 15036 mbflimlem 24270 climlimsup 42048 ioodvbdlimc1lem1 42223 |
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