How to prove a set is open

Hint: Let $S$ be the set of all $(x,y)$ such that $2\lt x^2+y^2\lt 4$. Let $(a,b)\in S$. We want to show that there is a positive $r$ such that the open disk with centre $(a,b)$ and radius $r$ is entirely contained in $S$.

Draw a picture. It is clear that if $r\le \min(\sqrt{a^2+b^2}-\sqrt{2}, \sqrt{4}-\sqrt{a^2+b^2})$ then the open disk with centre $(a,b)$ and radius $r$ is entirely contained in $S$.