# Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff

Proof attempt:

Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two disjoint open sets that separates $x,y$.

Let $B_\epsilon(x)$ and $B_\delta(y)$ be two metric balls containing $x,y$ respectively. Suppose that $B_\epsilon(x) \cap B_\delta(y) \neq \varnothing$

Stuck Here: Hmm...How should I adjust $\epsilon, \delta$ so that these balls are separated?

Idea: Reduce $\epsilon$ by half. If they are still intersecting...reduce $\delta$ by half. Continue ad infinitum

Is there a more satisfying solution i.e. closed form expression for reduced $\epsilon, \delta$ so they are no longer intersecting. Thanks!