A sequence in which liminf < limsup In this question we are looking for a sequence in which liminf is strictly less than limsup. A majority of the examples that I came up with and found were in which liminf<=limsup. As for the second part of this we are also looking for subsequences that converge to liminf and limsup. Any help is appreciated. 

Take $a_n=(-1)^n$. You have $$\liminf_{n\rightarrow\infty} (-1)^n = -1 < 1 = \limsup_{n\rightarrow\infty}(-1)^n$$
