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Theorem limsupbnd1 13285
Description: If a sequence is eventually at most  A, then the limsup is also at most  A. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence  1  /  n which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
Hypotheses
Ref Expression
limsupbnd.1  |-  ( ph  ->  B  C_  RR )
limsupbnd.2  |-  ( ph  ->  F : B --> RR* )
limsupbnd.3  |-  ( ph  ->  A  e.  RR* )
limsupbnd1.4  |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A ) )
Assertion
Ref Expression
limsupbnd1  |-  ( ph  ->  ( limsup `  F )  <_  A )
Distinct variable groups:    j, k, A    B, j, k    j, F, k    ph, j, k

Proof of Theorem limsupbnd1
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 limsupbnd1.4 . 2  |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A ) )
2 limsupbnd.1 . . . . . 6  |-  ( ph  ->  B  C_  RR )
32adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  B  C_  RR )
4 limsupbnd.2 . . . . . 6  |-  ( ph  ->  F : B --> RR* )
54adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  F : B
--> RR* )
6 simpr 461 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  k  e.  RR )
7 limsupbnd.3 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
87adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  A  e. 
RR* )
9 eqid 2467 . . . . . 6  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupgle 13280 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  k  e.  RR  /\  A  e.  RR* )  ->  ( ( ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  <->  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A
) ) )
113, 5, 6, 8, 10syl211anc 1234 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  <->  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A
) ) )
12 reex 9595 . . . . . . . . . . . 12  |-  RR  e.  _V
1312ssex 4597 . . . . . . . . . . 11  |-  ( B 
C_  RR  ->  B  e. 
_V )
142, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
15 xrex 11229 . . . . . . . . . . 11  |-  RR*  e.  _V
1615a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR*  e.  _V )
17 fex2 6750 . . . . . . . . . 10  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
184, 14, 16, 17syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
19 limsupcl 13276 . . . . . . . . 9  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
2018, 19syl 16 . . . . . . . 8  |-  ( ph  ->  ( limsup `  F )  e.  RR* )
21 xrleid 11368 . . . . . . . 8  |-  ( (
limsup `  F )  e. 
RR*  ->  ( limsup `  F
)  <_  ( limsup `  F ) )
2220, 21syl 16 . . . . . . 7  |-  ( ph  ->  ( limsup `  F )  <_  ( limsup `  F )
)
239limsuple 13281 . . . . . . . 8  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  ( limsup `  F )  e.  RR* )  ->  ( ( limsup `  F )  <_  ( limsup `
 F )  <->  A. k  e.  RR  ( limsup `  F
)  <_  ( (
n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) ) )
242, 4, 20, 23syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( limsup `  F
)  <_  ( limsup `  F )  <->  A. k  e.  RR  ( limsup `  F
)  <_  ( (
n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) ) )
2522, 24mpbid 210 . . . . . 6  |-  ( ph  ->  A. k  e.  RR  ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2625r19.21bi 2836 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  <_  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2720adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  e.  RR* )
289limsupgf 13278 . . . . . . . 8  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
2928a1i 11 . . . . . . 7  |-  ( ph  ->  ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR* )
3029ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
31 xrletr 11373 . . . . . 6  |-  ( ( ( limsup `  F )  e.  RR*  /\  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR*  /\  A  e. 
RR* )  ->  (
( ( limsup `  F
)  <_  ( (
n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  /\  ( ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3227, 30, 8, 31syl3anc 1228 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  /\  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3326, 32mpand 675 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  ->  ( limsup `
 F )  <_  A ) )
3411, 33sylbird 235 . . 3  |-  ( (
ph  /\  k  e.  RR )  ->  ( A. j  e.  B  (
k  <_  j  ->  ( F `  j )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3534rexlimdva 2959 . 2  |-  ( ph  ->  ( E. k  e.  RR  A. j  e.  B  ( k  <_ 
j  ->  ( F `  j )  <_  A
)  ->  ( limsup `  F )  <_  A
) )
361, 35mpd 15 1  |-  ( ph  ->  ( limsup `  F )  <_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   [,)cico 11543   limsupclsp 13273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-ico 11547  df-limsup 13274
This theorem is referenced by:  caucvgrlem  13475  limsupre  31506
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