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Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminflimsup | Structured version Visualization version GIF version |
Description: A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 42089). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminflimsup.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climliminflimsup.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climliminflimsup.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
Ref | Expression |
---|---|
climliminflimsup | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminflimsup.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climliminflimsup.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
4 | climliminflimsup.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
5 | 2, 1, 4 | climliminf 42107 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) |
6 | 5 | biimpd 231 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ → 𝐹 ⇝ (lim inf‘𝐹))) |
7 | 6 | imp 409 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ (lim inf‘𝐹)) |
8 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ) |
9 | 8 | ffvelrnda 6851 | . . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
10 | 1, 3, 7, 9 | climrecl 14940 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) ∈ ℝ) |
11 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
12 | 11 | limsupcld 41991 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) ∈ ℝ*) |
13 | 3, 1, 8, 11 | climliminflimsupd 42102 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
14 | 13 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
15 | 12, 14 | xreqled 41618 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) |
16 | 10, 15 | jca 514 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) |
17 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ) |
18 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ) |
19 | simprl 769 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ) | |
20 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
21 | 17, 1, 18, 19, 20 | liminflimsupclim 42108 | . 2 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ ) |
22 | 16, 21 | impbida 799 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 lim supclsp 14827 ⇝ cli 14841 lim infclsi 42052 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-ioo 12743 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-ceil 13164 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-liminf 42053 |
This theorem is referenced by: climliminflimsup2 42110 climliminflimsup3 42111 |
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