Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable

Let $(X,\mathcal{A})$ be a measurable space, let $d\ge 1$ be a natural number, and let $E$ be a nonempty subset of Euclidean space $\mathbb{R}^d$. Let $g:E\to\mathbb{R}$ be sequentially continuous on $E$: whenever $(x_n)_{n\in\mathbb{N}}$ is a sequence in $E$ and $x\in E$ with the Euclidean distance $d(x_n,x)$ converging to $0$, one has $g(x_n)\to g(x)$. Let $f=(f^1,\dots,f^d):X\to E$ be a map each of whose components $f^j:X\to\mathbb{R}$ is measurable with respect to $\mathcal{A}$ and the Borel $\sigma$-algebra on the real line.

Then the composition $g\circ f:X\to\mathbb{R}$ is measurable.