Independent Gaussian Random Variables are Jointly Gaussian

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $p$ be a natural number, and let $X_1,\dots,X_p$ be independent random variables on $(\Omega,\mathcal{F},P)$, each of which is a Gaussian random variable. Then $(X_1,\dots,X_p)$ is a Gaussian random vector, and its distinct components are uncorrelated: with the covariance of square-integrable random variables, defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector,