Weak Sequential Compactness of Borel Measures of Total Mass One on a Compact Metric Space

Let $(K,d)$ be a metric space, let $\mathcal{T}_d$ be the collection of subsets of $K$ that are open in $(K,d)$, which is a topology on $K$ by Metric Open Sets Form a Topology, and assume that $K$ is compact in $(K,\mathcal{T}_d)$.

Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, identities, additive inverses, multiplicative inverses and order of their ordered field structure; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line. Let $\mathbb{N}$ be the set of natural numbers.

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence whose terms are Borel measures on $(K,d)$, and assume that $\mu_n(K)=1$ for every $n\in\mathbb{N}$, so that each $\mu_n$ is finite.

Then there exist a sequence $(n_j)_{j\in\mathbb{N}}$ in $\mathbb{N}$ that is strictly increasing and a Borel measure $\mu$ on $(K,d)$ with $\mu(K)=1$ such that the subsequence $(\mu_{n_j})_{j\in\mathbb{N}}$ converges weakly to $\mu$.