VMO 2018 Problem 1

The sequence $(x_n)$ is defined as follows: $$x_1=2, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$ for all $n\geq 1$ . a. Prove that $(x_n)$ has a finite limit and find that limit. b. For every $n\geq 1$ , prove that $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$