Let $x_1>2$ and $x_{n+1} := 1 + \sqrt{x_n - 1}$ for all $n \in \mathbb{N}$. Show that $\{x_n\}$ is decreasing and bounded below, and find its limit.
I showed that it's bounded below ($x_n>2$ for all $n$) using induction. But then I don't know how to do the rest. Any help would be much appreciated.