P. 143 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
5.10  Elementary limits and convergence
5.10.1  Superior limit (lim sup)
Syntax	clsp 14201	Extend class notation to include the limsup function.
class limsup
Definition	df-limsup 14202*	Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 14205 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.)
|- limsup = ( x e. _V |-> inf ( ran ( k e. RR |-> sup ( ( ( x " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
Theorem	limsupgord 14203	Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> sup ( ( ( F " ( B [,) +oo ) ) i^i RR* ) , RR* , < ) <_ sup ( ( ( F " ( A [,) +oo ) ) i^i RR* ) , RR* , < ) )
Theorem	limsupcl 14204	Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.)
|- ( F e. V -> ( limsup ` F ) e. RR* )
Theorem	limsupval 14205*	The superior limit of an infinite sequence F of extended real numbers, which is the infimum of the set of suprema of all upper infinite subsequences of F. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2014.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- ( F e. V -> ( limsup ` F ) = inf ( ran G , RR* , < ) )
Theorem	limsupgf 14206*	Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- G : RR --> RR*
Theorem	limsupgval 14207*	Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- ( M e. RR -> ( G ` M ) = sup ( ( ( F " ( M [,) +oo ) ) i^i RR* ) , RR* , < ) )
Theorem	limsupgle 14208*	The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ C e. RR /\ A e. RR* ) -> ( ( G ` C ) <_ A <-> A. j e. B ( C <_ j -> ( F ` j ) <_ A ) ) )
Theorem	limsuple 14209*	The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( A <_ ( limsup ` F ) <-> A. j e. RR A <_ ( G ` j ) ) )
Theorem	limsuplt 14210*	The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   =>    |- ( ( B C_ RR /\ F : B --> RR* /\ A e. RR* ) -> ( ( limsup ` F ) < A <-> E. j e. RR ( G ` j ) < A ) )
Theorem	limsupval2 14211*	The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   &    |- ( ph -> F e. V )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> sup ( A , RR* , < ) = +oo )   =>    |- ( ph -> ( limsup ` F ) = inf ( ( G " A ) , RR* , < ) )
Theorem	limsupgre 14212*	If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- G = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )   &    |- Z = ( ZZ>= ` M )   =>    |- ( ( M e. ZZ /\ F : Z --> RR /\ ( limsup ` F ) < +oo ) -> G : RR --> RR )
Theorem	limsupbnd1 14213*	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.)
|- ( ph -> B C_ RR )   &    |- ( ph -> F : B --> RR* )   &    |- ( ph -> A e. RR* )   &    |- ( ph -> E. k e. RR A. j e. B ( k <_ j -> ( F ` j ) <_ A ) )   =>    |- ( ph -> ( limsup ` F ) <_ A )
Theorem	limsupbnd2 14214*	If a sequence is eventually greater than A, then the limsup is also greater than A. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- ( ph -> B C_ RR )   &    |- ( ph -> F : B --> RR* )   &    |- ( ph -> A e. RR* )   &    |- ( ph -> sup ( B , RR* , < ) = +oo )   &    |- ( ph -> E. k e. RR A. j e. B ( k <_ j -> A <_ ( F ` j ) ) )   =>    |- ( ph -> A <_ ( limsup ` F ) )
5.10.2  Limits
Syntax	cli 14215	Extend class notation with convergence relation for limits.
class ~~>
Syntax	crli 14216	Extend class notation with real convergence relation for limits.
class ~~> r
Syntax	co1 14217	Extend class notation with the set of all eventually bounded functions.
class O(1)
Syntax	clo1 14218	Extend class notation with the set of all eventually upper bounded functions.
class <_O(1)
Definition	df-clim 14219*	Define the limit relation for complex number sequences. See clim 14225 for its relational expression. (Contributed by NM, 28-Aug-2005.)
|- ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
Definition	df-rlim 14220*	Define the limit relation for partial functions on the reals. See rlim 14226 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ~~> r = { <. f , x >. | ( ( f e. ( CC ^pm RR ) /\ x e. CC ) /\ A. y e. RR+ E. z e. RR A. w e. dom f ( z <_ w -> ( abs ` ( ( f ` w ) - x ) ) < y ) ) }
Definition	df-o1 14221*	Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of O(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- O(1) = { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }
Definition	df-lo1 14222*	Define the set of eventually upper bounded real functions. This fills a gap in O(1) coverage, to express statements like f ( x ) <_ g ( x ) + O ( x ) via ( x e. RR+ |-> ( f ( x ) - g ( x ) ) / x ) e. <_O(1). (Contributed by Mario Carneiro, 25-May-2016.)
|- <_O(1) = { f e. ( RR ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( f ` y ) <_ m }
Theorem	climrel 14223	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Rel ~~>
Theorem	rlimrel 14224	The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
|- Rel ~~> r
Theorem	clim 14225*	Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ph -> F e. V )   &    |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	rlim 14226*	Express the predicate: The limit of complex number function F is C, or F converges to C, in the real sense. This means that for any real x, no matter how small, there always exists a number y such that the absolute difference of any number in the function beyond y and the limit is less than x. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ RR )   &    |- ( ( ph /\ z e. A ) -> ( F ` z ) = B )   =>    |- ( ph -> ( F ~~> r C <-> ( C e. CC /\ A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) ) ) )
Theorem	rlim2 14227*	Rewrite rlim 14226 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
|- ( ph -> A. z e. A B e. CC )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( z e. A |-> B ) ~~> r C <-> A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) ) )
Theorem	rlim2lt 14228*	Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
|- ( ph -> A. z e. A B e. CC )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( z e. A |-> B ) ~~> r C <-> A. x e. RR+ E. y e. RR A. z e. A ( y < z -> ( abs ` ( B - C ) ) < x ) ) )
Theorem	rlim3 14229*	Restrict the range of the domain bound to reals greater than some D e. RR. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( ph -> A. z e. A B e. CC )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. RR )   =>    |- ( ph -> ( ( z e. A |-> B ) ~~> r C <-> A. x e. RR+ E. y e. ( D [,) +oo ) A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) ) )
Theorem	climcl 14230	Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( F ~~> A -> A e. CC )
Theorem	rlimpm 14231	Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( F ~~> r A -> F e. ( CC ^pm RR ) )
Theorem	rlimf 14232	Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( F ~~> r A -> F : dom F --> CC )
Theorem	rlimss 14233	Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( F ~~> r A -> dom F C_ RR )
Theorem	rlimcl 14234	Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( F ~~> r A -> A e. CC )
Theorem	clim2 14235*	Express the predicate: The limit of complex number sequence F is A, or F converges to A, with more general quantifier restrictions than clim 14225. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	clim2c 14236*	Express the predicate F converges to A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )
Theorem	clim0 14237*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) )
Theorem	clim0c 14238*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
Theorem	rlim0 14239*	Express the predicate B ( z ) converges to 0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
|- ( ph -> A. z e. A B e. CC )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> ( ( z e. A |-> B ) ~~> r 0 <-> A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` B ) < x ) ) )
Theorem	rlim0lt 14240*	Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
|- ( ph -> A. z e. A B e. CC )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> ( ( z e. A |-> B ) ~~> r 0 <-> A. x e. RR+ E. y e. RR A. z e. A ( y < z -> ( abs ` B ) < x ) ) )
Theorem	climi 14241*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )
Theorem	climi2 14242*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )
Theorem	climi0 14243*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> 0 )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )
Theorem	rlimi 14244*	Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
|- ( ph -> A. z e. A B e. V )   &    |- ( ph -> R e. RR+ )   &    |- ( ph -> ( z e. A |-> B ) ~~> r C )   =>    |- ( ph -> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) )
Theorem	rlimi2 14245*	Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)
|- ( ph -> A. z e. A B e. V )   &    |- ( ph -> R e. RR+ )   &    |- ( ph -> ( z e. A |-> B ) ~~> r C )   &    |- ( ph -> D e. RR )   =>    |- ( ph -> E. y e. ( D [,) +oo ) A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) )
Theorem	ello1 14246*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( F e. <_O(1) <-> ( F e. ( RR ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( F ` y ) <_ m ) )
Theorem	ello12 14247*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( F : A --> RR /\ A C_ RR ) -> ( F e. <_O(1) <-> E. x e. RR E. m e. RR A. y e. A ( x <_ y -> ( F ` y ) <_ m ) ) )
Theorem	ello12r 14248*	Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( C e. RR /\ M e. RR ) /\ A. x e. A ( C <_ x -> ( F ` x ) <_ M ) ) -> F e. <_O(1) )
Theorem	lo1f 14249	An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( F e. <_O(1) -> F : dom F --> RR )
Theorem	lo1dm 14250	An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( F e. <_O(1) -> dom F C_ RR )
Theorem	lo1bdd 14251*	The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( F e. <_O(1) /\ F : A --> RR ) -> E. x e. RR E. m e. RR A. y e. A ( x <_ y -> ( F ` y ) <_ m ) )
Theorem	ello1mpt 14252*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   =>    |- ( ph -> ( ( x e. A |-> B ) e. <_O(1) <-> E. y e. RR E. m e. RR A. x e. A ( y <_ x -> B <_ m ) ) )
Theorem	ello1mpt2 14253*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( x e. A |-> B ) e. <_O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> B <_ m ) ) )
Theorem	ello1d 14254*	Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> M e. RR )   &    |- ( ( ph /\ ( x e. A /\ C <_ x ) ) -> B <_ M )   =>    |- ( ph -> ( x e. A |-> B ) e. <_O(1) )
Theorem	lo1bdd2 14255*	If an eventually bounded function is bounded on every interval A i^i ( -oo , y ) by a function M ( y ), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> ( x e. A |-> B ) e. <_O(1) )   &    |- ( ( ph /\ ( y e. RR /\ C <_ y ) ) -> M e. RR )   &    |- ( ( ( ph /\ x e. A ) /\ ( ( y e. RR /\ C <_ y ) /\ x < y ) ) -> B <_ M )   =>    |- ( ph -> E. m e. RR A. x e. A B <_ m )
Theorem	lo1bddrp 14256*	Refine o1bdd2 14272 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> ( x e. A |-> B ) e. <_O(1) )   &    |- ( ( ph /\ ( y e. RR /\ C <_ y ) ) -> M e. RR )   &    |- ( ( ( ph /\ x e. A ) /\ ( ( y e. RR /\ C <_ y ) /\ x < y ) ) -> B <_ M )   =>    |- ( ph -> E. m e. RR+ A. x e. A B <_ m )
Theorem	elo1 14257*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( F e. O(1) <-> ( F e. ( CC ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( abs ` ( F ` y ) ) <_ m ) )
Theorem	elo12 14258*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( F : A --> CC /\ A C_ RR ) -> ( F e. O(1) <-> E. x e. RR E. m e. RR A. y e. A ( x <_ y -> ( abs ` ( F ` y ) ) <_ m ) ) )
Theorem	elo12r 14259*	Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( ( F : A --> CC /\ A C_ RR ) /\ ( C e. RR /\ M e. RR ) /\ A. x e. A ( C <_ x -> ( abs ` ( F ` x ) ) <_ M ) ) -> F e. O(1) )
Theorem	o1f 14260	An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( F e. O(1) -> F : dom F --> CC )
Theorem	o1dm 14261	An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( F e. O(1) -> dom F C_ RR )
Theorem	o1bdd 14262*	The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( F e. O(1) /\ F : A --> CC ) -> E. x e. RR E. m e. RR A. y e. A ( x <_ y -> ( abs ` ( F ` y ) ) <_ m ) )
Theorem	lo1o1 14263	A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( F : A --> CC -> ( F e. O(1) <-> ( abs o. F ) e. <_O(1) ) )
Theorem	lo1o12 14264*	A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about <_O(1) to O(1).) (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
Theorem	elo1mpt 14265*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. RR E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )
Theorem	elo1mpt2 14266*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )
Theorem	elo1d 14267*	Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   &    |- ( ph -> C e. RR )   &    |- ( ph -> M e. RR )   &    |- ( ( ph /\ ( x e. A /\ C <_ x ) ) -> ( abs ` B ) <_ M )   =>    |- ( ph -> ( x e. A |-> B ) e. O(1) )
Theorem	o1lo1 14268*	A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)
|- ( ( ph /\ x e. A ) -> B e. RR )   =>    |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( ( x e. A |-> B ) e. <_O(1) /\ ( x e. A |-> -u B ) e. <_O(1) ) ) )
Theorem	o1lo12 14269*	A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> M e. RR )   &    |- ( ( ph /\ x e. A ) -> M <_ B )   =>    |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> B ) e. <_O(1) ) )
Theorem	o1lo1d 14270*	A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> ( x e. A |-> B ) e. O(1) )   =>    |- ( ph -> ( x e. A |-> B ) e. <_O(1) )
Theorem	icco1 14271*	Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)
|- ( ph -> A C_ RR )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> M e. RR )   &    |- ( ph -> N e. RR )   &    |- ( ( ph /\ ( x e. A /\ C <_ x ) ) -> B e. ( M [,] N ) )   =>    |- ( ph -> ( x e. A |-> B ) e. O(1) )
Theorem	o1bdd2 14272*	If an eventually bounded function is bounded on every interval A i^i ( -oo , y ) by a function M ( y ), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   &    |- ( ph -> ( x e. A |-> B ) e. O(1) )   &    |- ( ( ph /\ ( y e. RR /\ C <_ y ) ) -> M e. RR )   &    |- ( ( ( ph /\ x e. A ) /\ ( ( y e. RR /\ C <_ y ) /\ x < y ) ) -> ( abs ` B ) <_ M )   =>    |- ( ph -> E. m e. RR A. x e. A ( abs ` B ) <_ m )
Theorem	o1bddrp 14273*	Refine o1bdd2 14272 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   &    |- ( ph -> ( x e. A |-> B ) e. O(1) )   &    |- ( ( ph /\ ( y e. RR /\ C <_ y ) ) -> M e. RR )   &    |- ( ( ( ph /\ x e. A ) /\ ( ( y e. RR /\ C <_ y ) /\ x < y ) ) -> ( abs ` B ) <_ M )   =>    |- ( ph -> E. m e. RR+ A. x e. A ( abs ` B ) <_ m )
Theorem	climconst 14274*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   =>    |- ( ph -> F ~~> A )
Theorem	rlimconst 14275*	A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( ( A C_ RR /\ B e. CC ) -> ( x e. A |-> B ) ~~> r B )
Theorem	rlimclim1 14276	Forward direction of rlimclim 14277. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> r A )   &    |- ( ph -> Z C_ dom F )   =>    |- ( ph -> F ~~> A )
Theorem	rlimclim 14277	A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F : Z --> CC )   =>    |- ( ph -> ( F ~~> r A <-> F ~~> A ) )
Theorem	climrlim2 14278*	Produce a real limit from an integer limit, where the real function is only dependent on the integer part of x. (Contributed by Mario Carneiro, 2-May-2016.)
|- Z = ( ZZ>= ` M )   &    |- ( n = ( |_ ` x ) -> B = C )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> ( n e. Z |-> B ) ~~> D )   &    |- ( ( ph /\ n e. Z ) -> B e. CC )   &    |- ( ( ph /\ x e. A ) -> M <_ x )   =>    |- ( ph -> ( x e. A |-> C ) ~~> r D )
Theorem	climconst2 14279	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ>= ` M ) C_ Z   &    |- Z e. _V   =>    |- ( ( A e. CC /\ M e. ZZ ) -> ( Z X. { A } ) ~~> A )
Theorem	climz 14280	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ X. { 0 } ) ~~> 0
Theorem	rlimuni 14281	A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> sup ( A , RR* , < ) = +oo )   &    |- ( ph -> F ~~> r B )   &    |- ( ph -> F ~~> r C )   =>    |- ( ph -> B = C )
Theorem	rlimdm 14282	Two ways to express that a function has a limit. (The expression ( ~~> r ` F ) is sometimes useful as a shorthand for "the unique limit of the function F"). (Contributed by Mario Carneiro, 8-May-2016.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> sup ( A , RR* , < ) = +oo )   =>    |- ( ph -> ( F e. dom ~~> r <-> F ~~> r ( ~~> r ` F ) ) )
Theorem	climuni 14283	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- ( ( F ~~> A /\ F ~~> B ) -> A = B )
Theorem	fclim 14284	The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ~~> : dom ~~> --> CC
Theorem	climdm 14285	Two ways to express that a function has a limit. (The expression ( ~~> ` F ) is sometimes useful as a shorthand for "the unique limit of the function F"). (Contributed by Mario Carneiro, 18-Mar-2014.)
|- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
Theorem	climeu 14286*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
|- ( F ~~> A -> E! x F ~~> x )
Theorem	climreu 14287*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
|- ( F ~~> A -> E! x e. CC F ~~> x )
Theorem	climmo 14288*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
|- E* x F ~~> x
Theorem	rlimres 14289	The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( F ~~> r A -> ( F |` B ) ~~> r A )
Theorem	lo1res 14290	The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( F e. <_O(1) -> ( F |` A ) e. <_O(1) )
Theorem	o1res 14291	The restriction of an eventually bounded function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( F e. O(1) -> ( F |` A ) e. O(1) )
Theorem	rlimres2 14292*	The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( ph -> A C_ B )   &    |- ( ph -> ( x e. B |-> C ) ~~> r D )   =>    |- ( ph -> ( x e. A |-> C ) ~~> r D )
Theorem	lo1res2 14293*	The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ph -> A C_ B )   &    |- ( ph -> ( x e. B |-> C ) e. <_O(1) )   =>    |- ( ph -> ( x e. A |-> C ) e. <_O(1) )
Theorem	o1res2 14294*	The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.)
|- ( ph -> A C_ B )   &    |- ( ph -> ( x e. B |-> C ) e. O(1) )   =>    |- ( ph -> ( x e. A |-> C ) e. O(1) )
Theorem	lo1resb 14295	The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ph -> F : A --> RR )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( F e. <_O(1) <-> ( F |` ( B [,) +oo ) ) e. <_O(1) ) )
Theorem	rlimresb 14296	The restriction of a function to an unbounded-above interval converges iff the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( F ~~> r C <-> ( F |` ( B [,) +oo ) ) ~~> r C ) )
Theorem	o1resb 14297	The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( F e. O(1) <-> ( F |` ( B [,) +oo ) ) e. O(1) ) )
Theorem	climeq 14298*	Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( F ~~> A <-> G ~~> A ) )
Theorem	lo1eq 14299*	Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ( ph /\ x e. A ) -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )   =>    |- ( ph -> ( ( x e. A |-> B ) e. <_O(1) <-> ( x e. A |-> C ) e. <_O(1) ) )
Theorem	rlimeq 14300*	Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( ( ph /\ x e. A ) -> B e. CC )   &    |- ( ( ph /\ x e. A ) -> C e. CC )   &    |- ( ph -> D e. RR )   &    |- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )   =>    |- ( ph -> ( ( x e. A |-> B ) ~~> r E <-> ( x e. A |-> C ) ~~> r E ) )