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Theorem caurcvg 12472
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 Mario Carneiro, 8-May-2016.)
Hypotheses
Ref Expression
caurcvg.1  |-  Z  =  ( ZZ>= `  M )
caurcvg.3  |-  ( ph  ->  F : Z --> RR )
caurcvg.4  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
Assertion
Ref Expression
caurcvg  |-  ( ph  ->  F  ~~>  ( limsup `  F
) )
Distinct variable groups:    k, m, x, F    m, M, x    ph, k, m, x    k, Z, m, x
Allowed substitution hint:    M( k)

Proof of Theorem caurcvg
StepHypRef Expression
1 caurcvg.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
2 uzssz 10507 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3380 . . . . 5  |-  Z  C_  ZZ
4 zssre 10291 . . . . 5  |-  ZZ  C_  RR
53, 4sstri 3359 . . . 4  |-  Z  C_  RR
65a1i 11 . . 3  |-  ( ph  ->  Z  C_  RR )
7 caurcvg.3 . . 3  |-  ( ph  ->  F : Z --> RR )
8 1rp 10618 . . . . . 6  |-  1  e.  RR+
9 ne0i 3636 . . . . . 6  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
108, 9ax-mp 8 . . . . 5  |-  RR+  =/=  (/)
11 caurcvg.4 . . . . 5  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
12 r19.2z 3719 . . . . 5  |-  ( (
RR+  =/=  (/)  /\  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
1310, 11, 12sylancr 646 . . . 4  |-  ( ph  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
14 eluzel2 10495 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1514, 1eleq2s 2530 . . . . . . . 8  |-  ( m  e.  Z  ->  M  e.  ZZ )
161uzsup 11246 . . . . . . . 8  |-  ( M  e.  ZZ  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1715, 16syl 16 . . . . . . 7  |-  ( m  e.  Z  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1817a1d 24 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  sup ( Z ,  RR* ,  <  )  =  +oo ) )
1918rexlimiv 2826 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
2019rexlimivw 2828 . . . 4  |-  ( E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
2113, 20syl 16 . . 3  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
223sseli 3346 . . . . . . . . . . . 12  |-  ( m  e.  Z  ->  m  e.  ZZ )
233sseli 3346 . . . . . . . . . . . 12  |-  ( k  e.  Z  ->  k  e.  ZZ )
24 eluz 10501 . . . . . . . . . . . 12  |-  ( ( m  e.  ZZ  /\  k  e.  ZZ )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2522, 23, 24syl2an 465 . . . . . . . . . . 11  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2625biimprd 216 . . . . . . . . . 10  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( m  <_  k  ->  k  e.  ( ZZ>= `  m ) ) )
2726expimpd 588 . . . . . . . . 9  |-  ( m  e.  Z  ->  (
( k  e.  Z  /\  m  <_  k )  ->  k  e.  (
ZZ>= `  m ) ) )
2827imim1d 72 . . . . . . . 8  |-  ( m  e.  Z  ->  (
( k  e.  (
ZZ>= `  m )  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  ( ( k  e.  Z  /\  m  <_  k )  ->  ( abs `  ( ( F `
 k )  -  ( F `  m ) ) )  <  x
) ) )
2928exp4a 591 . . . . . . 7  |-  ( m  e.  Z  ->  (
( k  e.  (
ZZ>= `  m )  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  ( k  e.  Z  ->  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) ) ) )
3029ralimdv2 2788 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  A. k  e.  Z  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) ) )
3130reximia 2813 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  E. m  e.  Z  A. k  e.  Z  ( m  <_  k  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x ) )
3231ralimi 2783 . . . 4  |-  ( A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  Z  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) )
3311, 32syl 16 . . 3  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  Z  (
m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
346, 7, 21, 33caurcvgr 12469 . 2  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
3515a1d 24 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  M  e.  ZZ ) )
3635rexlimiv 2826 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  M  e.  ZZ )
3736rexlimivw 2828 . . . 4  |-  ( E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  M  e.  ZZ )
3813, 37syl 16 . . 3  |-  ( ph  ->  M  e.  ZZ )
39 ax-resscn 9049 . . . 4  |-  RR  C_  CC
40 fss 5601 . . . 4  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
417, 39, 40sylancl 645 . . 3  |-  ( ph  ->  F : Z --> CC )
421, 38, 41rlimclim 12342 . 2  |-  ( ph  ->  ( F  ~~> r  (
limsup `  F )  <->  F  ~~>  ( limsup `  F ) ) )
4334, 42mpbid 203 1  |-  ( ph  ->  F  ~~>  ( limsup `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4214   -->wf 5452   ` cfv 5456  (class class class)co 6083   supcsup 7447   CCcc 8990   RRcr 8991   1c1 8993    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   abscabs 12041   limsupclsp 12266    ~~> cli 12280    ~~> r crli 12281
This theorem is referenced by:  caurcvg2  12473  mbflimlem  19561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fl 11204  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285
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