Let $X_n, X$ take values in $\mathcal{X} \subseteq \mathbb{R}^d$ and let $g : \mathcal{X} \to \mathbb{R}$ be continuous. Then \begin{align*} X_n \xrightarrow{a.s./\mathbb{P}/d} X \implies g(X_n) \xrightarrow{a.s./\mathbb{P}/d} g(X). \end{align*} That is, if $X_n$ converges to $X$ in any of the three modes, then $g(X_n)$ converges to $g(X)$ in the same mode.