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Let $\{x_n\}_{n=1}^\infty$ be a sequence of points in $\mathbb R$. Let $X$ be a set defined as a collection of all points in the sequence $\{x_n\}_{n=1}^{\infty}$.

Is the following claim true?

$\left\{x_n\right\}_{n=1}^\infty$ converges to a limit $x^*$ if and only if the set $X$ has a limit point.

My intuition is that the claim is true but I'm not quite sure how to go about showing a rigorous proof of it.

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    Think about what happens if the points in the set are isolated from each other (such as will be the case for a finite set), but the sequence is such that a tail of the sequence is just one value is repeated over and over again.2011-09-16

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