A computer program ouputs the digits of $\pi$ by evaluating the recurrence relation

$a_{n+1} = a_n + sin \ a_n$

with $a_0 = \frac{6}{5}$

Does the sequence actually converge or is this just coincidence?

I would like to answer this by rewriting it as a differential equation.

Can I write $\frac{dy}{dx} = sin \ x$ and solve this?