Show that for almost all $x$ in $[-1,1]$, the series $ \sum\limits_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $ converges

Question

Let $\{r_n\}$ be a sequence of real numbers in $[-1, 1]$, then show that for almost all $x$ in $[-1,1]$, the series $$ \sum_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $$ converges.

I am struggling on this problem of real analysis. I know that somehow I have to use the monotone convergence theorem. But I cannot see it clearly. How to get started?