I have a question of finding lim sup and lim inf of $a_n=\frac{1}{n} + (-1)^n$ and prove $\liminf a_n \leq \limsup a_n.$ So the work below is what I did for the first part.
$a_{odd\ n} = \frac{1}{n}-1$ and $a_{even\ n} = \frac{1}{n}+1.$ So $\limsup a_n = 1$ and $\liminf a_n = -1.$ How do I prove the second part?? I tried to use the definition but I am confused with the definition.