# Limit Point is Limit of Convergent Sequence/Proof 2

## Theorem

Let $M = \left({X, d}\right)$ be a metric space.

Let $E \subseteq X$ be a subset of $X$.

Let $p$ be a limit point of $E$.

Then there exists a sequence $\left\langle{x_n}\right\rangle \subseteq E$ which converges to a limit:

$\displaystyle \lim_{n \mathop \to \infty} x_n = p$

where $\displaystyle \lim_{n \mathop \to \infty} x_n$ is the limit of the sequence $\left\langle{x_n}\right\rangle$.

## Proof

Then Equivalence of Definitions of Limit Point can be applied.

$\blacksquare$