Show that is monotone.
I'm getting trouble to do that. It's a part of another problem which says to prove that converges, so in order to do that, I already proved that is bounded below.
This particular sequence may or may not be monotonic. The given equation may have 2 solutions (depending on b), and therefore the given sequence isn't well defined - it could take on one of these values, or bounce between the 2. In any case, it is more likely that the problem has a typo.