AUSTRIAN-POLISH 1994 Problem 2

The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$ , $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$ , $c_{n+1}=c_n^2-2c_n+2$ Prove that for all $n\ge 1$ , $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$