Problem

Source: Vietnam MO 2018 1st day 1st problem

Tags: calculus, limit, Sequence



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 $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$