Completeness of the Quadratic Wasserstein Space over Euclidean Space

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ be the Wasserstein space with its distance, a metric space by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric.

(Completeness)¶ $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ is a complete metric space: every Cauchy sequence in it converges to an element of $\mathcal{P}_{2}(\mathbb{R}^{d})$.