Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, and let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence of Borel measures on $(X,d)$ with $\mu_{n}(X)=1$ for every $n\in\mathbb{N}$ that is tight in $(X,d)$.

¶ Then there are a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and a Borel measure $\mu$ on $(X,d)$ with $\mu(X)=1$ such that $(\mu_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $\mu$.