VMO 2015 Problem 1

Given a non negative real $a$ and a sequence $(u_n)$ defined by $$\begin{cases} u_1=3 u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases}$$

a) Prove that for $a=0$ , the sequence is convergent and find its limit.

b) For $a\in [0,1]$ , prove that the sequence if convergent.