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

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$ and let $d_{E}$ be the Euclidean distance on $\mathbb{R}^{m}$, so that $(\mathbb{R}^{m},d_{E})$ is a metric space whose Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{m})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, and each $\mu\in\mathcal{P}(\mathbb{R}^{m})$ is a Borel measure on it with $\mu(\mathbb{R}^{m})=1$.

Let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence whose terms belong to $\mathcal{P}(\mathbb{R}^{m})$, and assume that it is tight in $(\mathbb{R}^{m},d_{E})$.

Then there exist a sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ that is strictly increasing and a measure $\mu\in\mathcal{P}(\mathbb{R}^{m})$ such that the subsequence $(\mu_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $\mu$ on the metric space $(\mathbb{R}^{m},d_{E})$.