Finding disjoint neighborhoods of two points in $\Bbb R$ Let $x$ and $y$ be unique real numbers. How do you prove that there exists a neighborhood $P$ of $x$ and a neighborhood $Q$ of $y$ such that $P \cap Q = \emptyset$?

Hint: open intervals of length $e$ centered at $x$ and $y$ are such neighborhoods, and  if $e$ is small enough, they will not intersect.  How small does $e$ have to be for this to happen?
