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. 

How about 
$$a_n=
\begin{cases}
1+\frac1n, & n\text{ even}\\
-1-\frac1n & n\text{ odd}.
\end{cases}$$
