Every sequence in a compact space is a lingering sequence/Statement

Statement

In a metric space $(X,d)$ that is compact every sequence is a lingering sequence, that is to say:

• $$\forall(x_n)_{n=1}^\infty\subseteq X\ \exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0]$$