Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $\mathcal{P}_{2}(X)$ be the set of probability measures with finite second moment and $W_{2}$ the quadratic Wasserstein distance, so that $(\mathcal{P}_{2}(X),W_{2})$ is a metric space by The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §metric. Then the following hold.

1. (Cauchy sequences)¶ Every Cauchy sequence $(\mu_{m})_{m\in\mathbb{N}}$ in $(\mathcal{P}_{2}(X),W_{2})$ is tight in $(X,d)$.

2. (Weakly convergent sequences)¶ Every sequence $(\mu_{j})_{j\in\mathbb{N}}$ in $\mathcal{P}(X)$ that converges weakly to some $\mu\in\mathcal{P}(X)$, $\mu_{j}\Rightarrow\mu$ in the sense of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §weak, is tight in $(X,d)$.