Upper and lower limits of 'alternating' sequence

Question

Suppose $(a_n)$ is a real sequence and $a_1,a_3,a_5,\ldots$ converges to $s$, and $a_2,a_4,a_6,\ldots$ converges to $i$ where $s\ge i$, where $s=+\infty$ and $i=-\infty$ is a possibility, then is it always true that $\limsup_\limits{n\to \infty} a_n=s$ and $\liminf_\limits{n\to \infty} a_n=i$ ? If yes, then why?