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Definition df-limsup 13537
Description: Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 13542 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.)
Assertion
Ref Expression
df-limsup  |-  limsup  =  ( x  e.  _V  |-> inf ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  <  ) )
Distinct variable group:    x, k

Detailed syntax breakdown of Definition df-limsup
StepHypRef Expression
1 clsp 13535 . 2  class  limsup
2 vx . . 3  setvar  x
3 cvv 3013 . . 3  class  _V
4 vk . . . . . 6  setvar  k
5 cr 9525 . . . . . 6  class  RR
62cv 1447 . . . . . . . . 9  class  x
74cv 1447 . . . . . . . . . 10  class  k
8 cpnf 9659 . . . . . . . . . 10  class +oo
9 cico 11627 . . . . . . . . . 10  class  [,)
107, 8, 9co 6276 . . . . . . . . 9  class  ( k [,) +oo )
116, 10cima 4815 . . . . . . . 8  class  ( x
" ( k [,) +oo ) )
12 cxr 9661 . . . . . . . 8  class  RR*
1311, 12cin 3371 . . . . . . 7  class  ( ( x " ( k [,) +oo ) )  i^i  RR* )
14 clt 9662 . . . . . . 7  class  <
1513, 12, 14csup 7941 . . . . . 6  class  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )
164, 5, 15cmpt 4433 . . . . 5  class  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1716crn 4813 . . . 4  class  ran  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1817, 12, 14cinf 7942 . . 3  class inf ( ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  <  )
192, 3, 18cmpt 4433 . 2  class  ( x  e.  _V  |-> inf ( ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  <  ) )
201, 19wceq 1448 1  wff  limsup  =  ( x  e.  _V  |-> inf ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  <  ) )
Colors of variables: wff setvar class
This definition is referenced by:  limsupcl  13540  limsupval  13542
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