Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a complete and separable metric space, let $\mathcal{T}_{d}$ be the collection of subsets of $X$ that are open in $(X,d)$, a topology on $X$ by Metric Open Sets Form a Topology, and let $\mu$ be a Borel measure on $(X,d)$ with $\mu(X)<\infty$.

¶ Then the set $\{\mu\}$ is tight in $(X,d)$: for every real $\varepsilon>0$ there is a set $K\subseteq X$, compact in $(X,\mathcal{T}_{d})$, with $\mu(X\setminus K)\le\varepsilon$.