Show that $\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}},\cdots$

Question

Show that $$\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}},\cdots$$ is increasing.

Thanks for any help.

Answer 1

We have $a_n = \sqrt{2+a_{n-1}}$. Inductive step: $$a_n> a_{n-1} \implies 2+a_n > 2+ a_{n-1} \implies \sqrt{2+a_n} > \sqrt{2 + a_{n-1}} \implies a_{n+1} > a_n$$

We were allowed to take the square root because clearly $a_n>0$ for all $n$.

Answer 2

Hint: The function $\sqrt{x}$ is increasing