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Theorem limsupbnd1 13487
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 AV, 12-Sep-2020.)
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 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  B  C_  RR )
4 limsupbnd.2 . . . . . 6  |-  ( ph  ->  F : B --> RR* )
54adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  F : B
--> RR* )
6 simpr 462 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  k  e.  RR )
7 limsupbnd.3 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
87adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  A  e. 
RR* )
9 eqid 2428 . . . . . 6  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupgle 13478 . . . . 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 1270 . . . 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 9581 . . . . . . . . . . . 12  |-  RR  e.  _V
1312ssex 4511 . . . . . . . . . . 11  |-  ( B 
C_  RR  ->  B  e. 
_V )
142, 13syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
15 xrex 11250 . . . . . . . . . . 11  |-  RR*  e.  _V
1615a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR*  e.  _V )
17 fex2 6706 . . . . . . . . . 10  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
184, 14, 16, 17syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
19 limsupcl 13472 . . . . . . . . 9  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
2018, 19syl 17 . . . . . . . 8  |-  ( ph  ->  ( limsup `  F )  e.  RR* )
21 xrleid 11400 . . . . . . . 8  |-  ( (
limsup `  F )  e. 
RR*  ->  ( limsup `  F
)  <_  ( limsup `  F ) )
2220, 21syl 17 . . . . . . 7  |-  ( ph  ->  ( limsup `  F )  <_  ( limsup `  F )
)
239limsuple 13479 . . . . . . . 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 1264 . . . . . . 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 213 . . . . . 6  |-  ( ph  ->  A. k  e.  RR  ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2625r19.21bi 2734 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  <_  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2720adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  e.  RR* )
289limsupgf 13476 . . . . . . . 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 5981 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
31 xrletr 11406 . . . . . 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 1264 . . . . 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 679 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  ->  ( limsup `
 F )  <_  A ) )
3411, 33sylbird 238 . . 3  |-  ( (
ph  /\  k  e.  RR )  ->  ( A. j  e.  B  (
k  <_  j  ->  ( F `  j )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3534rexlimdva 2856 . 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 187    /\ wa 370    e. wcel 1872   A.wral 2714   E.wrex 2715   _Vcvv 3022    i^i cin 3378    C_ wss 3379   class class class wbr 4366    |-> cmpt 4425   "cima 4799   -->wf 5540   ` cfv 5544  (class class class)co 6249   supcsup 7907   RRcr 9489   +oocpnf 9623   RR*cxr 9625    < clt 9626    <_ cle 9627   [,)cico 11588   limsupclsp 13467
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 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-ico 11592  df-limsup 13469
This theorem is referenced by:  caucvgrlem  13679  limsupre  37604
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