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Theorem caurcvgr 13737
Description: A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that  F is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.) (Revised by AV, 12-Sep-2020.)
Hypotheses
Ref Expression
caurcvgr.1  |-  ( ph  ->  A  C_  RR )
caurcvgr.2  |-  ( ph  ->  F : A --> RR )
caurcvgr.3  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
caurcvgr.4  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) )
Assertion
Ref Expression
caurcvgr  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Distinct variable groups:    j, k, x, A    j, F, k, x    ph, j, k, x

Proof of Theorem caurcvgr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 caurcvgr.1 . . . . 5  |-  ( ph  ->  A  C_  RR )
2 caurcvgr.2 . . . . 5  |-  ( ph  ->  F : A --> RR )
3 caurcvgr.3 . . . . 5  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
4 caurcvgr.4 . . . . 5  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) )
5 1rp 11313 . . . . . 6  |-  1  e.  RR+
65a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR+ )
71, 2, 3, 4, 6caucvgrlem 13735 . . . 4  |-  ( ph  ->  E. j  e.  A  ( ( limsup `  F
)  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  1 ) ) ) )
8 simpl 458 . . . . 5  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
98rexlimivw 2911 . . . 4  |-  ( E. j  e.  A  ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
107, 9syl 17 . . 3  |-  ( ph  ->  ( limsup `  F )  e.  RR )
1110recnd 9676 . 2  |-  ( ph  ->  ( limsup `  F )  e.  CC )
121adantr 466 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
132adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : A
--> RR )
143adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( A ,  RR* ,  <  )  = +oo )
154adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( F `  j )
) )  <  x
) )
16 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
17 3re 10690 . . . . . . . . 9  |-  3  e.  RR
18 3pos 10710 . . . . . . . . 9  |-  0  <  3
1917, 18elrpii 11312 . . . . . . . 8  |-  3  e.  RR+
20 rpdivcl 11332 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  3  e.  RR+ )  ->  (
y  /  3 )  e.  RR+ )
2116, 19, 20sylancl 666 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  /  3 )  e.  RR+ )
2212, 13, 14, 15, 21caucvgrlem 13735 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  A  ( ( limsup `
 F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) ) )
23 simpr 462 . . . . . . 7  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )  ->  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
2423reximi 2890 . . . . . 6  |-  ( E. j  e.  A  ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )  ->  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
2522, 24syl 17 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
26 ssrexv 3526 . . . . 5  |-  ( A 
C_  RR  ->  ( E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  (
y  /  3 ) ) )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) ) )
2712, 25, 26sylc 62 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
28 rpcn 11317 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e.  CC )
2928adantl 467 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  CC )
30 3cn 10691 . . . . . . . . 9  |-  3  e.  CC
3130a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  e.  CC )
32 3ne0 10711 . . . . . . . . 9  |-  3  =/=  0
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  =/=  0 )
3429, 31, 33divcan2d 10392 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( 3  x.  ( y  / 
3 ) )  =  y )
3534breq2d 4435 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( abs `  ( ( F `
 k )  -  ( limsup `  F )
) )  <  (
3  x.  ( y  /  3 ) )  <-> 
( abs `  (
( F `  k
)  -  ( limsup `  F ) ) )  <  y ) )
3635imbi2d 317 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  (
y  /  3 ) ) )  <->  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) ) )
3736rexralbidv 2944 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) )  <->  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
y ) ) )
3827, 37mpbid 213 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) )
3938ralrimiva 2836 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  RR  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
y ) )
40 ax-resscn 9603 . . . 4  |-  RR  C_  CC
41 fss 5754 . . . 4  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
422, 40, 41sylancl 666 . . 3  |-  ( ph  ->  F : A --> CC )
43 eqidd 2423 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  =  ( F `  k ) )
4442, 1, 43rlim 13558 . 2  |-  ( ph  ->  ( F  ~~> r  (
limsup `  F )  <->  ( ( limsup `
 F )  e.  CC  /\  A. y  e.  RR+  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) ) ) )
4511, 39, 44mpbir2and 930 1  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772    C_ wss 3436   class class class wbr 4423   -->wf 5597   ` cfv 5601  (class class class)co 6305   supcsup 7963   CCcc 9544   RRcr 9545   0cc0 9546   1c1 9547    x. cmul 9551   +oocpnf 9679   RR*cxr 9681    < clt 9682    <_ cle 9683    - cmin 9867    / cdiv 10276   3c3 10667   RR+crp 11309   abscabs 13297   limsupclsp 13523    ~~> r crli 13548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13525  df-rlim 13552
This theorem is referenced by:  caucvgrlem2  13739  caurcvg  13741
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