Let $(X,d_{X})$ be a metric space, and for every integer $n\geq 1$, let $f_{n}:X\to\textbf{R}$ be a real-valued function. Suppose that $f_{n}$ converges pointwise to another function $f:X\to\textbf{R}$ on $X$ (in this question we give $\textbf{R}$ the standard metric $d(x,y) = |x-y|$). Let $h:\textbf{R}\to\textbf{R}$ be a continuous function. Show that the functions $h\circ f_{n}$ converge pointwise to $h\circ f$ on $X$, where $h\circ f_{n}:X\to\textbf{R}$ is the function $h\circ f_{n}(x) = h(f_{n}(x))$, and similarly for $h\circ f$.
MY ATTEMPT
My argument is the following: since for each fixed $x\in X$, the sequence $f_{n}(x)$ is a convergent sequence of real numbers which converges to $f(x)$, and continuous functions map convergent sequences onto convergent sequences, we conclude that $h(f_{n}(x))\to h(f(x))$, and we are done.
Am I missing any formal step or my argument is fine?