![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupgt | Structured version Visualization version GIF version |
Description: Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupgt.k | β’ β²ππΉ |
limsupgt.m | β’ (π β π β β€) |
limsupgt.z | β’ π = (β€β₯βπ) |
limsupgt.f | β’ (π β πΉ:πβΆβ) |
limsupgt.r | β’ (π β (lim supβπΉ) β β) |
limsupgt.x | β’ (π β π β β+) |
Ref | Expression |
---|---|
limsupgt | β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupgt.m | . . 3 β’ (π β π β β€) | |
2 | limsupgt.z | . . 3 β’ π = (β€β₯βπ) | |
3 | limsupgt.f | . . 3 β’ (π β πΉ:πβΆβ) | |
4 | limsupgt.r | . . 3 β’ (π β (lim supβπΉ) β β) | |
5 | limsupgt.x | . . 3 β’ (π β π β β+) | |
6 | 1, 2, 3, 4, 5 | limsupgtlem 45212 | . 2 β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
7 | limsupgt.k | . . . . . . . . 9 β’ β²ππΉ | |
8 | nfcv 2899 | . . . . . . . . 9 β’ β²ππ | |
9 | 7, 8 | nffv 6912 | . . . . . . . 8 β’ β²π(πΉβπ) |
10 | nfcv 2899 | . . . . . . . 8 β’ β²π β | |
11 | nfcv 2899 | . . . . . . . 8 β’ β²ππ | |
12 | 9, 10, 11 | nfov 7456 | . . . . . . 7 β’ β²π((πΉβπ) β π) |
13 | nfcv 2899 | . . . . . . 7 β’ β²π < | |
14 | nfcv 2899 | . . . . . . . 8 β’ β²πlim sup | |
15 | 14, 7 | nffv 6912 | . . . . . . 7 β’ β²π(lim supβπΉ) |
16 | 12, 13, 15 | nfbr 5199 | . . . . . 6 β’ β²π((πΉβπ) β π) < (lim supβπΉ) |
17 | nfv 1909 | . . . . . 6 β’ β²π((πΉβπ) β π) < (lim supβπΉ) | |
18 | fveq2 6902 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
19 | 18 | oveq1d 7441 | . . . . . . 7 β’ (π = π β ((πΉβπ) β π) = ((πΉβπ) β π)) |
20 | 19 | breq1d 5162 | . . . . . 6 β’ (π = π β (((πΉβπ) β π) < (lim supβπΉ) β ((πΉβπ) β π) < (lim supβπΉ))) |
21 | 16, 17, 20 | cbvralw 3301 | . . . . 5 β’ (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
22 | 21 | a1i 11 | . . . 4 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
23 | fveq2 6902 | . . . . 5 β’ (π = π β (β€β₯βπ) = (β€β₯βπ)) | |
24 | 23 | raleqdv 3323 | . . . 4 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
25 | 22, 24 | bitrd 278 | . . 3 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
26 | 25 | cbvrexvw 3233 | . 2 β’ (βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
27 | 6, 26 | sylib 217 | 1 β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β²wnfc 2879 βwral 3058 βwrex 3067 class class class wbr 5152 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcr 11147 < clt 11288 β cmin 11484 β€cz 12598 β€β₯cuz 12862 β+crp 13016 lim supclsp 15456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-xadd 13135 df-ico 13372 df-fz 13527 df-fzo 13670 df-fl 13799 df-ceil 13800 df-limsup 15457 |
This theorem is referenced by: liminfltlem 45239 liminflimsupclim 45242 |
Copyright terms: Public domain | W3C validator |