Space is both $T_1$-space and normal space is Hausdorff space

Question

Every metric space which is both $T_1$-space and normal space is Hausdorff space.

$(X,\tau)$ is Hausdorff if $\forall x,y\in X\exists U,V\in\tau$ such that $x\in U$ and $y\in V$ and $U\cap V=\emptyset$.

It easy to note that every Hasudorff space is $T_1$-space. However I know that every metrizable space is Hausdorff therefore it is $T_1$-space. However I cannot necessarily see or coneceive a proof that a metric space which is both $T_1$-space and normal space is Hausdorff space.

Question:

Can anyone help me provide a proof?

Thanks in advance!

Answer 1

Hint: In a $T_1$ space points are closed.