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 ) |
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 ) |