Let $S$ be a non empty subset of $\Bbb R$ and bounded above and $M=\sup S$.Then prove that either $M \in S$ or $M$ is a limit point of $S$.

I tried the math but I am stuck at the point that how to show that $M \in S$. I did the part that $M$ is a limit point of $S$.$M=SupS$.Let $M$ does not belong to $S$. $M=sup s$So $x<M$ for all x belongs to S. And for any epsilon>0,there exists a y belongs to S such that (M- epsilon)<y<M implies that (M-epsilon)<y<(M+epsilon) .Implies that M is a limit point of S.