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Theorem caurcvg 12114
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 10214 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3183 . . . . 5  |-  Z  C_  ZZ
4 zssre 9998 . . . . 5  |-  ZZ  C_  RR
53, 4sstri 3163 . . . 4  |-  Z  C_  RR
65a1i 12 . . 3  |-  ( ph  ->  Z  C_  RR )
7 caurcvg.3 . . 3  |-  ( ph  ->  F : Z --> RR )
8 1rp 10325 . . . . . 6  |-  1  e.  RR+
9 ne0i 3436 . . . . . 6  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
108, 9ax-mp 10 . . . . 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 3518 . . . . 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 647 . . . 4  |-  ( ph  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
14 eluzel2 10202 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1514, 1eleq2s 2350 . . . . . . . 8  |-  ( m  e.  Z  ->  M  e.  ZZ )
161uzsup 10933 . . . . . . . 8  |-  ( M  e.  ZZ  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1715, 16syl 17 . . . . . . 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 2636 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
2019rexlimivw 2638 . . . 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 17 . . 3  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
223sseli 3151 . . . . . . . . . . . 12  |-  ( m  e.  Z  ->  m  e.  ZZ )
233sseli 3151 . . . . . . . . . . . 12  |-  ( k  e.  Z  ->  k  e.  ZZ )
24 eluz 10208 . . . . . . . . . . . 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 589 . . . . . . . . 9  |-  ( m  e.  Z  ->  (
( k  e.  Z  /\  m  <_  k )  ->  k  e.  (
ZZ>= `  m ) ) )
2827imim1d 71 . . . . . . . 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 592 . . . . . . 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 2598 . . . . . 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 2623 . . . . 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 2593 . . . 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 17 . . 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 12111 . 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 2636 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  M  e.  ZZ )
3736rexlimivw 2638 . . . 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 17 . . 3  |-  ( ph  ->  M  e.  ZZ )
39 ax-resscn 8762 . . . 4  |-  RR  C_  CC
40 fss 5335 . . . 4  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
417, 39, 40sylancl 646 . . 3  |-  ( ph  ->  F : Z --> CC )
421, 38, 41rlimclim 11985 . 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519    C_ wss 3127   (/)c0 3430   class class class wbr 3997   -->wf 4669   ` cfv 4673  (class class class)co 5792   supcsup 7161   CCcc 8703   RRcr 8704   1c1 8706    +oocpnf 8832   RR*cxr 8834    < clt 8835    <_ cle 8836    - cmin 9005   ZZcz 9991   ZZ>=cuz 10197   RR+crp 10321   abscabs 11684   limsupclsp 11909    ~~> cli 11923    ~~> r crli 11924
This theorem is referenced by:  caurcvg2  12115  mbflimlem  18984
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-ico 10628  df-fl 10891  df-seq 11013  df-exp 11071  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928
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