Three parts to this question:

Consider the sequence {$\displaystyle {a_n}$}, where $\displaystyle a_n = \sqrt{2+\sqrt{2+...+\sqrt{2}}} $

Use Induction to prove that {$\displaystyle {a_n}$} is an increasing sequence, that is $\displaystyle a_{n+1} > a_n$ for all n = 1, 2, 3...

Use Induction to prove that {$\displaystyle a_n$} is bounded from above by 2, that is, $\displaystyle a_n < 2$ for all n = 1, 2, 3...

Is {$\displaystyle a_n$} convergent?