Show that if $\lim(x_n - x_{n-2}) = 0$ then $\lim(\frac{x_n - x_{n-1}}{n}) = 0$

Question

I know that $\frac{1}{n}$ converges to zero. So if I can show that $x_n - x_{n-1}$ is bounded then by the squeeze theorem, I would have the desired result. I have tried to show that $x_n - x_{n-1}$ is bounded by seeing $(x_n - x_{n-2}$ as a Cauchy sequence but I am not getting anywhere.