Mean-Square Limits of Gaussian Random Variables are Gaussian

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $(X_k)_{k\in\mathbb{N}}$ be a sequence of Gaussian random variables on it, and let $X$ be a square-integrable random variable such that the mean-square distance satisfies $\lVert X_k-X\rVert_{2}\to0$ as $k\to\infty$.

Then $X$ is a Gaussian random variable, and its expectation and variance are the limits

where each $\mathbb{E}[X_k]$ and $\operatorname{Var}(X_k)$ is defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector.