MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caurcvg Unicode version

Theorem caurcvg 12165
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 10263 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3221 . . . . 5  |-  Z  C_  ZZ
4 zssre 10047 . . . . 5  |-  ZZ  C_  RR
53, 4sstri 3201 . . . 4  |-  Z  C_  RR
65a1i 10 . . 3  |-  ( ph  ->  Z  C_  RR )
7 caurcvg.3 . . 3  |-  ( ph  ->  F : Z --> RR )
8 1rp 10374 . . . . . 6  |-  1  e.  RR+
9 ne0i 3474 . . . . . 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 3556 . . . . 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 644 . . . 4  |-  ( ph  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
14 eluzel2 10251 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1514, 1eleq2s 2388 . . . . . . . 8  |-  ( m  e.  Z  ->  M  e.  ZZ )
161uzsup 10983 . . . . . . . 8  |-  ( M  e.  ZZ  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1715, 16syl 15 . . . . . . 7  |-  ( m  e.  Z  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1817a1d 22 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  sup ( Z ,  RR* ,  <  )  =  +oo ) )
1918rexlimiv 2674 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
2019rexlimivw 2676 . . . 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 15 . . 3  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
223sseli 3189 . . . . . . . . . . . 12  |-  ( m  e.  Z  ->  m  e.  ZZ )
233sseli 3189 . . . . . . . . . . . 12  |-  ( k  e.  Z  ->  k  e.  ZZ )
24 eluz 10257 . . . . . . . . . . . 12  |-  ( ( m  e.  ZZ  /\  k  e.  ZZ )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2522, 23, 24syl2an 463 . . . . . . . . . . 11  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2625biimprd 214 . . . . . . . . . 10  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( m  <_  k  ->  k  e.  ( ZZ>= `  m ) ) )
2726expimpd 586 . . . . . . . . 9  |-  ( m  e.  Z  ->  (
( k  e.  Z  /\  m  <_  k )  ->  k  e.  (
ZZ>= `  m ) ) )
2827imim1d 69 . . . . . . . 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 589 . . . . . . 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 2636 . . . . . 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 2661 . . . . 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 2631 . . . 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 15 . . 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 12162 . 2  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
3515a1d 22 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  M  e.  ZZ ) )
3635rexlimiv 2674 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  M  e.  ZZ )
3736rexlimivw 2676 . . . 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 15 . . 3  |-  ( ph  ->  M  e.  ZZ )
39 ax-resscn 8810 . . . 4  |-  RR  C_  CC
40 fss 5413 . . . 4  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
417, 39, 40sylancl 643 . . 3  |-  ( ph  ->  F : Z --> CC )
421, 38, 41rlimclim 12036 . 2  |-  ( ph  ->  ( F  ~~> r  (
limsup `  F )  <->  F  ~~>  ( limsup `  F ) ) )
4334, 42mpbid 201 1  |-  ( ph  ->  F  ~~>  ( limsup `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   1c1 8754    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   abscabs 11735   limsupclsp 11960    ~~> cli 11974    ~~> r crli 11975
This theorem is referenced by:  caurcvg2  12166  mbflimlem  19038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979
  Copyright terms: Public domain W3C validator