real analysis - A convergent sequence that is defined recursively

I would like to get a hint on how to establish the convergence of the following sequence:

$$a_{n+1}= a_n + \frac{\sqrt{\vert a_n \vert }}{n^2}$$

where $a_1$ is arbitrary. This is an increasing sequence, so if I could show that it was bounded above I would be done. I cannot figure out how to do that. Any help would be appreciated.

Answer

We have $a_n=\sum_{j=1}^{n-1}\frac{\sqrt{|a_j|}}{j^2}+a_1$. Let $M$ such that $M>|a_1|+\frac{\pi^2}6\sqrt M$. Assume that $|a_j|\leqslant M$ for all $1\leqslant j\leqslant n-1$. Then
$$|a_n|\leqslant \sqrt M\sum_{j=1}^{n-1}\frac 1{j^2}+|a_1|As $|a_1|