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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 43784). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminflimsup.1 | β’ (π β π β β€) |
climliminflimsup.2 | β’ π = (β€β₯βπ) |
climliminflimsup.3 | β’ (π β πΉ:πβΆβ) |
Ref | Expression |
---|---|
climliminflimsup | β’ (π β (πΉ β dom β β ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminflimsup.2 | . . . 4 β’ π = (β€β₯βπ) | |
2 | climliminflimsup.1 | . . . . 5 β’ (π β π β β€) | |
3 | 2 | adantr 482 | . . . 4 β’ ((π β§ πΉ β dom β ) β π β β€) |
4 | climliminflimsup.3 | . . . . . . 7 β’ (π β πΉ:πβΆβ) | |
5 | 2, 1, 4 | climliminf 43802 | . . . . . 6 β’ (π β (πΉ β dom β β πΉ β (lim infβπΉ))) |
6 | 5 | biimpd 228 | . . . . 5 β’ (π β (πΉ β dom β β πΉ β (lim infβπΉ))) |
7 | 6 | imp 408 | . . . 4 β’ ((π β§ πΉ β dom β ) β πΉ β (lim infβπΉ)) |
8 | 4 | adantr 482 | . . . . 5 β’ ((π β§ πΉ β dom β ) β πΉ:πβΆβ) |
9 | 8 | ffvelcdmda 7030 | . . . 4 β’ (((π β§ πΉ β dom β ) β§ π β π) β (πΉβπ) β β) |
10 | 1, 3, 7, 9 | climrecl 15400 | . . 3 β’ ((π β§ πΉ β dom β ) β (lim infβπΉ) β β) |
11 | simpr 486 | . . . . 5 β’ ((π β§ πΉ β dom β ) β πΉ β dom β ) | |
12 | 11 | limsupcld 43686 | . . . 4 β’ ((π β§ πΉ β dom β ) β (lim supβπΉ) β β*) |
13 | 3, 1, 8, 11 | climliminflimsupd 43797 | . . . . 5 β’ ((π β§ πΉ β dom β ) β (lim infβπΉ) = (lim supβπΉ)) |
14 | 13 | eqcomd 2744 | . . . 4 β’ ((π β§ πΉ β dom β ) β (lim supβπΉ) = (lim infβπΉ)) |
15 | 12, 14 | xreqled 43323 | . . 3 β’ ((π β§ πΉ β dom β ) β (lim supβπΉ) β€ (lim infβπΉ)) |
16 | 10, 15 | jca 513 | . 2 β’ ((π β§ πΉ β dom β ) β ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) |
17 | 2 | adantr 482 | . . 3 β’ ((π β§ ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) β π β β€) |
18 | 4 | adantr 482 | . . 3 β’ ((π β§ ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) β πΉ:πβΆβ) |
19 | simprl 770 | . . 3 β’ ((π β§ ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) β (lim infβπΉ) β β) | |
20 | simprr 772 | . . 3 β’ ((π β§ ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) β (lim supβπΉ) β€ (lim infβπΉ)) | |
21 | 17, 1, 18, 19, 20 | liminflimsupclim 43803 | . 2 β’ ((π β§ ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ))) β πΉ β dom β ) |
22 | 16, 21 | impbida 800 | 1 β’ (π β (πΉ β dom β β ((lim infβπΉ) β β β§ (lim supβπΉ) β€ (lim infβπΉ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5104 dom cdm 5631 βΆwf 6488 βcfv 6492 βcr 10984 β€ cle 11124 β€cz 12433 β€β₯cuz 12696 lim supclsp 15287 β cli 15301 lim infclsi 43747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-pm 8702 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-q 12803 df-rp 12845 df-xneg 12962 df-xadd 12963 df-ioo 13197 df-ico 13199 df-fz 13354 df-fzo 13497 df-fl 13626 df-ceil 13627 df-seq 13836 df-exp 13897 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-limsup 15288 df-clim 15305 df-rlim 15306 df-liminf 43748 |
This theorem is referenced by: climliminflimsup2 43805 climliminflimsup3 43806 |
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