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.

If you already prove that $x_n> 2$, then  $$\dfrac{x_{n+1}-1}{x_n -1} = \dfrac{1}{\sqrt{x_n -1}} < 1 $$
$$\implies x_{n+1} -1 < x_n -1$$
So $x_n$ is decreasing and bounded below. The limit point is already in the other answers.
