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 ∈ ℕ ↦ if ( 2 ∥ 𝑛 , 0 , 1 ) )
Assertion	liminfltlimsupex	⊢ ( lim inf ‘ 𝐹 ) < ( lim sup ‘ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		liminfltlimsupex.1	⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 2 ∥ 𝑛 , 0 , 1 ) )
2		0lt1	⊢ 0 < 1
3	1	liminf10ex	⊢ ( lim inf ‘ 𝐹 ) = 0
4	1	limsup10ex	⊢ ( lim sup ‘ 𝐹 ) = 1
5	3 4	breq12i	⊢ ( ( lim inf ‘ 𝐹 ) < ( lim sup ‘ 𝐹 ) ↔ 0 < 1 )
6	2 5	mpbir	⊢ ( lim inf ‘ 𝐹 ) < ( lim sup ‘ 𝐹 )