I would like to find out if my proof is alright or if it is missing any steps...thanks.
$B$ is a non-empty open set. Show that every point in $B$ is a limit point of $B$.
Since $B$ is open, $\forall b \in B$, there exists an $\epsilon$-neighborhood $V_\epsilon (b) \subseteq B$. This means that for all $b \in B$, every single possible $\epsilon$-neighborhood of b intersects $B$ at some point other than $b$. By definition, this means that all elements of B are limit points of B.