Accumulation Points Are The Same As Limit Points

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Problem 51: (Accumulation Points Are The Same As Limit Points)

Let $S$ be a set. Prove that $a$ is an accumulation point of $S$ if and only if $a$ is a limit point of $S$. In other words, prove both of the following:

• If $a$ is an accumulation point of $S$, then $a$ is a limit point of $S$.
• If $a$ is a limit point of $S$, then $a$ is an accumulation point of $S$.

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