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Theorem limsupbnd1 12276
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 452 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  B  C_  RR )
4 limsupbnd.2 . . . . . 6  |-  ( ph  ->  F : B --> RR* )
54adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  F : B
--> RR* )
6 simpr 448 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  k  e.  RR )
7 limsupbnd.3 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
87adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  A  e. 
RR* )
9 eqid 2436 . . . . . 6  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( n  e.  RR  |->  sup (
( ( F "
( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupgle 12271 . . . . 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 1190 . . . 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 9081 . . . . . . . . . . . 12  |-  RR  e.  _V
1312ssex 4347 . . . . . . . . . . 11  |-  ( B 
C_  RR  ->  B  e. 
_V )
142, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
15 xrex 10609 . . . . . . . . . . 11  |-  RR*  e.  _V
1615a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR*  e.  _V )
17 fex2 5603 . . . . . . . . . 10  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
184, 14, 16, 17syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
19 limsupcl 12267 . . . . . . . . 9  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
2018, 19syl 16 . . . . . . . 8  |-  ( ph  ->  ( limsup `  F )  e.  RR* )
21 xrleid 10743 . . . . . . . 8  |-  ( (
limsup `  F )  e. 
RR*  ->  ( limsup `  F
)  <_  ( limsup `  F ) )
2220, 21syl 16 . . . . . . 7  |-  ( ph  ->  ( limsup `  F )  <_  ( limsup `  F )
)
239limsuple 12272 . . . . . . . 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 1184 . . . . . . 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 202 . . . . . 6  |-  ( ph  ->  A. k  e.  RR  ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2625r19.21bi 2804 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  <_  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2720adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  e.  RR* )
289limsupgf 12269 . . . . . . . 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 5870 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
31 xrletr 10748 . . . . . 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 1184 . . . . 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 657 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  ->  ( limsup `
 F )  <_  A ) )
3411, 33sylbird 227 . . 3  |-  ( (
ph  /\  k  e.  RR )  ->  ( A. j  e.  B  (
k  <_  j  ->  ( F `  j )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3534rexlimdva 2830 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    i^i cin 3319    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121   [,)cico 10918   limsupclsp 12264
This theorem is referenced by:  caucvgrlem  12466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-ico 10922  df-limsup 12265
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