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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.

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    What is your definition of "boundary point"?2011-10-23
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    If not, the segment $[p,q]$ would be partitioned by the two sets of points respectively interior and exterior to $S$. What's insufferable about that?2011-10-23
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    @ChrisEagle: I would say $x$ is a boundary point of $S$ if every open neighborhood of $x$ intersects both $S$ and the complement of $S$.2011-10-23

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