## [SOLVED] Limit of a sequence - recursion

I have this problem:

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

$
a_0=c$

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

I believe it's needed to prove that for every n : $a_n

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