$(10$ points$)$ We define the sequence $(s_n)$ recursively by letting $s_1=1$ and $s_{n+1}=\sqrt[\phantom{2}]{2s_n+2}.$

$($a$)$ Prove that $(s_n)$ is convergent.

$($b$)$ Find $\underset{?}{lim}{s}_n.$

If you could do this on paper it would be appreciated.