How to prove a limit is irrational

Question

There is a sequence $x_n$ such that $x_n^2 - 7 \le \frac 1{16} (x_{n-1}^2-7)^2$

How does this show the limit of $x_n$ is irrational?

(Also, $2\le x_n \le \frac 72$ and each $x_n$ is rational)

Thank you!