Prove the positive sequence $a_{n+1} = \sqrt{1+\frac{a^2_n}{4}} $ is strictly increasing for $0 \leq a_0<\frac{2}{\sqrt{3}}$ My attempt: 
$a_{n+1} - a_n=  \sqrt{1+\frac{a^2_n}{4}} -a_n > \frac{a_n}{2}-a_n = -\frac{1}{2}a_n$, which doesn't tell me any thing. How do I prove that this sequence is strickly increasing? Thank you! 

HINT: 
It remains to show that
$$
\forall n\, a_n<\frac2{\sqrt{3}}.
$$
For $n=1$ it is in comments.
