Let $S$ be a subset of the metric space $E$. A point $p \in S$ is called interior point of $S$ if there is an open ball in $E$ of center $p$ which is contained in $S$. Prove that the set of interior points of $S$ is an open subset of $E$ (called the interior of $S$).
It seems as if the statement has already proven what it wants me to show. In order to prove that something is an open set I need to find an open ball centered at some $p$ with radius $r$. If the set of open balls is contained in subset $S$ then $S$ is open. Yet I feel that I'm oversimplifying the actual process of proving this.
Anyone have any idea on this?