Independence is Preserved by Limits in Probability

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $p\ge1$ be a natural number, and for each $k\in\mathbb{N}$ let $X^{k}_{1},\dots,X^{k}_{p}$ be independent random variables on it. Let $X_1,\dots,X_p$ be random variables such that for each $i\in\{1,\dots,p\}$ the sequence $(X^{k}_{i})_{k}$ converges to $X_i$ in probability as $k\to\infty$.

Then $X_1,\dots,X_p$ are independent.