Given the positive sequence $a_{n+2} = \sqrt{a_{n+1}}+ \sqrt{a_n}$,
I want to prove these.
1) $|a_{n+2}| > 1 $ for sufficiently large $n \ge N$.
2) Let $b_{n} = |a_{n} - 4|$. Show that $b_{n+2} < (b_{n+1} + b_{n})/3$ for $n \ge N$.
3) Prove that the sequence converges.
How should I proceed? Is there a recurrence formula for $a_{n}$ like a continued fraction?