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Metamath Proof Explorer

Theorem liminflelimsupcex

Description: A counterexample for liminflelimsup , showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022)

		Ref	Expression
	Assertion	liminflelimsupcex	$⊢ lim sup (\emptyset) < lim inf (\emptyset)$

Step	Hyp	Ref	Expression
1		mnfltpnf	$⊢ −∞ < +∞$
2		limsup0	$⊢ lim sup (\emptyset) = −∞$
3		liminf0	$⊢ lim inf (\emptyset) = +∞$
4	2 3	breq12i	$⊢ lim sup (\emptyset) < lim inf (\emptyset) \leftrightarrow −∞ < +∞$
5	1 4	mpbir	$⊢ lim sup (\emptyset) < lim inf (\emptyset)$