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

From Maths

## Statement

In a metric space [ilmath](X,d)[/ilmath] that is compact every sequence is a lingering sequence, that is to say^{[1]}:

- [math]\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][/math]