Metric is jointly continuous

Statement

Let $(X,d)$ be a metric space. Then $X$ is also a topological space in the induced topology, and we can consider the metric as a map of topological spaces $d:X\times X\to \mathbb {R}$. This map is jointly continuous, i.e. it is continuous from $X\times X$ given the product topology.

Proof

It suffices to show that inverse images of open subsets of the form $(-\infty ,a)$ and $(b,\infty )$ are open subsets of $X\times X$. We will use the triangle inequality to prove this.