real analysis - Show that the recursive sequence converges and find its limit

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.

Answer

Any recursively defined sequence where the recursive relation (in this case $1+\sqrt{x_n - 1}$) is continuous can only ever converge to $x$ which are stationary points, i.e. points so that if you enter $x$ into the relation, you get $x$ back.

This means that if your sequence converges to some $x$, then it must be the case that $$x=1+\sqrt{x-1}$$