Theorem liminfltlimsupex

Description: An example where the liminf is strictly smaller than the limsup . (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	liminfltlimsupex.1	|- F = ( n e. NN |-> if ( 2 || n , 0 , 1 ) )
Assertion	liminfltlimsupex	|- ( liminf ` F ) < ( limsup ` F )

Proof

Step	Hyp	Ref	Expression
1		liminfltlimsupex.1	|- F = ( n e. NN |-> if ( 2 || n , 0 , 1 ) )
2		0lt1	|- 0 < 1
3	1	liminf10ex	|- ( liminf ` F ) = 0
4	1	limsup10ex	|- ( limsup ` F ) = 1
5	3 4	breq12i	|- ( ( liminf ` F ) < ( limsup ` F ) <-> 0 < 1 )
6	2 5	mpbir	|- ( liminf ` F ) < ( limsup ` F )