I'm taking second year Calc in my university, and we were told to prove this:
Let $S$ be an open set in $\mathbb{R}^n$ and $p \in S$ and $q \notin S$. Prove that a boundary point of $S$ is on the line segment joining $p$ and $q$.
I know that it's obvious, but can't seem to actually prove this result. I'm in a somewhat basic course, so I know a few things: connectedness and disconnectedness, how to use balls in an open set, what an accumulation point is, but some of the more rigorous topological terms might be unfamiliar.