INMO 2012 Problem 3

Define a sequence $\left\langle f_{0}(x), f_{1}(x), f_{2}(x), \ldots\right\rangle$ of functions by

$$f_{0}(x)=1, f_{1}(x)=x, \left(f_{n}(x)\right)^{2}-1=f_{n+1}(x) f_{n-1}(x), \text { for } n \geq 1$$

Prove that each $f_{n}(x)$ is a polynomial with integer coefficients.