> [!definition] > > Let $A \subseteq M$ where $M$ is a [[Metric Space|metric space]]. A point $c \in M$ is called a *cluster* point (limit point) if for any > $ > \forall \varepsilon > 0, \exists x \in A, x \ne c: \abs{x - c} < \varepsilon > $ > [!theorem] > > Let $A \subseteq M$ where $M$ is a [[Metric Space|metric space]] and $c \in M$. The following are equivalent: > 1. $c$ is a cluster point of $A$ ([[Topological Space|Topology]]). > 2. $\exists (x_n), x_n \in A, x_n \ne c \forall n \in \nat: \lim_{n \to \infty}x_n = c$ ([[Sequence]]). > > *Proof*. Let $c$ be a cluster point of $A$, and take $\varepsilon = \frac{1}{n}$. Since > $ > \forall \varepsilon > 0, \exists x_n \in A, x_n \ne c: |x_n - c| < \varepsilon = \frac{1}{n} > $ > A sequence may be constructed. > > Let $(x_n)$ be a [[Sequence|sequence]], $x_n \in A, x_n \ne c \forall n \in \nat$ with [[Limit|limit]] $\lim_{n \to \infty}x_n = c$. Since > $ > \forall \varepsilon > 0, \exists N \in \nat: \forall n \ge N, |x_n - c| < \varepsilon > $ > $c$ is a cluster point of $A$.