How to prove there's an upper bound on a series

Question

There's a series as following. $$ \sqrt{3}, \sqrt{3 + \sqrt{3}}, \sqrt{3 + \sqrt{3 + \sqrt{3}}} + \cdots $$ Is there any way I can find the upper bound to prove the limit exist? I've tried to write down the formula of this series as following: $$ x_n = \sqrt{3 + x_{n - 1}} $$ but still find no way to modify it to get to the upper bound.

Answer 1

Hint : Show inductively that all the terms are bounded by $$\frac{1+\sqrt{13}}{2}$$