Let $\displaystyle x$ and $\displaystyle y$ be distinct real numbers. Prove that there is a neighborhood $\displaystyle P$ of $\displaystyle x$ and a neighborhood $\displaystyle Q$ of $\displaystyle y$ such that $\displaystyle P \cap Q = \emptyset$.

The book left this proof as an exercise and the instructor said nothing about it. I'm just curious as to what the proof might be.