I have this problem:

Let 0<c<1, prove that the following sequence has a limit, and find it:

$\displaystyle a_n=\frac{c}{2}+\frac{a^2_{n-1}}{2}$

I believe it's needed to prove that for every n : $\displaystyle a_n<a_{n-1}$

I tried to prove it by induction, without success - I got a false phrase in the end of it ($\displaystyle a_n>1$ for every $\displaystyle n>1$, which is not neccesarily true - look at $\displaystyle a_2$)