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.