Is it true that every finite subset of a Hausdorff space has no limit point?

Question

Let $X$ be a Hausdorff space. Assume a finite $A\subset X$ has a limit point $b$, say. Every pair $a\in A, b\in X$ has disjoint neighborhoods. Denote those neighborhoods of $b$ by $U_a$ for each $a\in A$. Now, $\cap U_a$ is clearly a neighborhood of $b$ and disjoint from $A$, contradicting the fact that $b$ is a limit point of $A$. So I think the answer is yes, if I didn't make any mistakes.