I am considering a contractive sequence of the form
$$x_{n+1} = (1-ax_n)x_n$$
where $x_0\in[0,1/a)$. This sequence is obviously decreasing
$$x_{n+1} < x_n$$
but I would like an upper bound on the convergence rate. And I have no idea how to approach this problem nor what keywords to google about it, so any clue/direction would be helpful.
I have also reason to believe (from the context this sequence stems from) that the convergence speed is probably something like $O(1/n)$ but do not have any proof for that other than that I get similar rates if I tackle the problem from a different angle.