The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, let $d_{E}$ denote the Euclidean distance on $\mathbb{R}^{m+m}$, and let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{m})$, with $\Pi(\mu,\nu)$ the set of their couplings.

Let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence with $\pi_{n}\in\Pi(\mu,\nu)$ for every $n\in\mathbb{N}$, and assume that it converges weakly to $\pi\in\mathcal{P}(\mathbb{R}^{m+m})$ on the metric space $(\mathbb{R}^{m+m},d_{E})$.

Then $\pi\in\Pi(\mu,\nu)$.