PAMO 2023 Problem 3

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \text{for all } n \geq 1. \end{align*} Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$ . (South Africa)