You can't assume convergence if you're trying to prove it...
@moemoe: The usual way to prove that this type of recursive sequences converge is to prove by induction that the sequence is monotone and bounded.
In your case, plug in the values for the first few terms to see whether the sequence is increasing or decreasing, and then prove it by induction. After you do that, prove that it is bounded (this can be done by induction as well).
EDIT: Plato ahead of me :<
The sequence is defined recursively as...
(1)
The function is represented here...
It has a single zero at and because is less that the line crossing the x axis in with unity negative slope, any will produce a sequence converging at without oscillations...
Kind regards