I have this problem:

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

<br />
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<a_{n-1}

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)