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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 11583 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 3598 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
4 | zssre 11261 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
5 | 3, 4 | sstri 3577 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ) |
7 | caurcvg.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
8 | 1rp 11712 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
9 | 8 | ne0ii 3882 | . . . . 5 ⊢ ℝ+ ≠ ∅ |
10 | caurcvg.4 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) | |
11 | r19.2z 4012 | . . . . 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 694 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) |
13 | eluzel2 11568 | . . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
14 | 13, 1 | eleq2s 2706 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ) |
15 | 1 | uzsup 12524 | . . . . . . . 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 3009 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → sup(𝑍, ℝ*, < ) = +∞) |
19 | 18 | rexlimivw 3011 | . . . 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 3564 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
22 | 3 | sseli 3564 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
23 | eluz 11577 | . . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) | |
24 | 21, 22, 23 | syl2an 493 | . . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑚) ↔ 𝑚 ≤ 𝑘)) |
25 | 24 | biimprd 237 | . . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑚 ≤ 𝑘 → 𝑘 ∈ (ℤ≥‘𝑚))) |
26 | 25 | expimpd 627 | . . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ∧ 𝑚 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑚))) |
27 | 26 | imim1d 80 | . . . . . . . 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 631 | . . . . . . 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 2944 | . . . . . 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 2992 | . . . . 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 2936 | . . . 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 14252 | . 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 3009 | . . . . 5 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 → 𝑀 ∈ ℤ) |
36 | 35 | rexlimivw 3011 | . . . 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 9872 | . . . 4 ⊢ ℝ ⊆ ℂ | |
39 | fss 5969 | . . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑍⟶ℂ) | |
40 | 7, 38, 39 | sylancl 693 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
41 | 1, 37, 40 | rlimclim 14125 | . 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 (lim sup‘𝐹) ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
42 | 33, 41 | mpbid 221 | 1 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℂcc 9813 ℝcr 9814 1c1 9816 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ℝ+crp 11708 abscabs 13822 lim supclsp 14049 ⇝ cli 14063 ⇝𝑟 crli 14064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 |
This theorem is referenced by: caurcvg2 14256 mbflimlem 23240 ioodvbdlimc1lem1 38821 |
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