Infinite matrix such that $|a_{i,j}| \leq r^{|i-j|}$

Question

Let $A=(a_{i,j})_{i,j \in \mathbb{N}}$ be an infinite matrix.

Assume there exists $r < 1$ such that $$|a_{i,j}| \leq r^{|i-j|} , \forall i,j \in \mathbb{N}$$

Show that A define a bounded operator on $l^2(\mathbb{N})$ with $$||A|| \leq \frac{1+r}{1-r}$$.

I've used Cauchy inequality to get $$||Ax|| \leq ||x|| \sum_{i,j}|a_{i,j}|^2$$ but after some calculus, i've find that ht $\sum_{i,j}r^{2|i-j|}$ is divergent.

I think I should use the test Schur test to get the majoration of the norme while proving that $A$ is bounded or maybe i could use a Cauchy Swart inequality more "refined" to proove this but I don't find one.

May I have a hint ?

Thanks !