Regular implies semiregular

Statement
Any regular space is a semiregular space.

Proof
Given: A regular space $$X$$. A point $$x \in X$$ and any open subset $$V \subseteq X$$ containing $$x$$,

To prove: There exists a regular open subset $$U$$ of $$X$$ containing $$x$$ and contained in $$V$$.

Proof: