ZA-SAMO 2000 Problem 3

Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by $$\begin{aligned} a_1 & = 2, a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned}$$ Prove that $a_n$ is an integer for all $n$ .