Metric Space is First-Countable

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

Then $M$ is first-countable.

Proof
Let $x \in X$.

Let:
 * $\mathcal B = \left\{{B_{1/n} \left({x}\right): n \in \N_{>0}}\right\}$

where $B_{\epsilon} \left({x}\right)$ denotes the open $\epsilon$-ball of $x$ in $M$.

By Surjection from Natural Numbers iff Countable, we have that $\mathcal B$ is countable.

By the definition of a first-countable space, it suffices to show that $\mathcal B$ is a local basis at $x$.

We have that an open ball is open.

Let $U$ be an open neighborhood of $x$.

By the definition of an open set, there exists a strictly positive real number $\epsilon$ such that $B_{\epsilon} \left({x}\right) \subseteq U$.

By the Archimedean principle, there exists a natural number $\displaystyle n > \frac 1 \epsilon$.

That is, $\displaystyle \frac 1 n < \epsilon$, and so $B_{1/n} \left({x}\right) \subseteq B_{\epsilon} \left({x}\right) \subseteq U$.

The result follows because $\subseteq$ is a transitive relation.