Lingering sequence

Given a metric space $(X,d)$ a lingering sequence or sometimes hovering sequence is a sequence $(x_n)_{n=1}^\infty\subseteq X$ that satisfies the following property:

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

Or in words:

• This

Theorems

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• Every convergent sequence is a lingering sequence

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Every convergent sequence is a lingering sequence/Statement

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• Every lingering sequence has a convergent subsequence

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Let $(X,d)$ be a metric space, then:

• $$\forall(x_n)_{n=1}^\infty\subseteq X\left[\left(\exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0]\right)\implies\left(\exists(k_n)_{n=1}^\infty\subseteq\mathbb{N}\left[(\forall n\in\mathbb{N}[k_n<k_{n+1}])\implies\left(\exists x'\in X\left[\lim_{n\rightarrow\infty}(x_{k_n})=x'\right]\right)\right]\right)\right]$$

This is just a verbose way of expressing the statement that:

• Given a sequence $(x_n)_{n=1}^\infty\subseteq X$ if it is a lingering sequence then it has a subsequence that converges

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• Every sequence in a compact space is a lingering sequence

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

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