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Not closet set

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Grigorios Kostakos
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Not closet set

Let $(X,\rho)$ a metric space and $(x_n)_{n\in{\mathbb{N}}}$ a Cauchy sequence in $X$, such that the set $\{x_n\;|\; n\in{\mathbb{N}}\}$ of the terms of this sequence it isn't a closed set. Prove that exists $x\in X$, such that $x_n\stackrel{\rho}{\longrightarrow}x$.
Grigorios Kostakos

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