All of the proofs I read of this are pretty much the same, but they all don't explain the last part of the proof and end it there because I assume they think it is obvious to everyone.
The proofs usually go like: Let $S = \bigcup_{i \in I} S_i$ for open sets $S_i$. Prove that $S$ is open. If $x$ is in some $S_i$, since $S_i$ is open, $\forall x \in S_i$ $\exists r > 0$ such that $N_r(x) \subset S_i \subset \bigcup\limits_{i \in I} S_i = S$. Thus $S$ is open.
I don't understand the final implication. We want to show that in $S$, every element of all sets are possible to be the center of an open ball around them with some radius that is contained within $S$. We've only stated that some $S_i$ is an open subset of $S$, which is actually a given so we haven't proven anything. Can someone explain the final part?