Limit of Sequence to Zero Distance Point

Theorem
Let $S$ be a non-empty subset of $\R$.

Suppose the distance $d \left({\xi, S}\right) = 0$ for some $\xi \in \R$.

Then there exists a sequence $\left \langle {x_n} \right \rangle$ in $S$ such that $\displaystyle \lim_{n \to \infty} x_n = \xi$.

Proof
First it is shown that:
 * $\forall n \in \N_{>0}: \exists x_n \in S: \left|{\xi - x_n}\right| < \dfrac 1 n$.

Suppose the contrary: that:
 * $\exists n \in \N_{>0}: \not \exists x \in S: \left|{\xi - x}\right| < \dfrac 1 n$

Then $\dfrac 1 n$ is a lower bound of the set $T = \left\{{\left|{\xi - x}\right|: x \in S}\right\}$.

This contradicts the assertion that $d \left({\xi, S}\right) = 0$.

We have from Sequence of Powers of Reciprocals is Null Sequence that:
 * $\displaystyle \lim_{n \to \infty} \dfrac 1 n = 0$

So as $\left|{\xi - x_n}\right| < \dfrac 1 n$ it follows from the Squeeze Theorem for Real Sequences that:
 * $\displaystyle \lim_{n \to \infty} x_n = \xi$

Hence the result.