All Limit Points Of An Open Interval

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Problem 6: (All Limit Points Of An Open Interval)

Let $S=(4,7)$ which is the open interval from 4 to 7 that does not include the end points. The previous problem's proof shows that $7$ is a limit point of $S$, and by symmetry $4$ is a limit point as well. Prove that if $p\in (4,7)$, then $p$ is a limit point of $S$. Then prove that if $p>7$ or $p<4$, then $p$ is not a limit point of $S$.

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