Let $X\subset \mathbb{R}^d$ and $A\subset \mathbb{R}$ be a compact sets, where $C_B(X)$ and $C_B(A)$ denote the space of continuous and bounded real-valued functions $X\rightarrow\mathbb{R}$ and $A\rightarrow\mathbb{R}$ respectively. Define $\mathcal{F}\subset C_B(A)$, and $\mathcal{G}\subset C(X)$ to be compact sets with respect to the metric induced by the norm $||f||_\infty =\sup_{x\in X}|f(x)|$. Then define $\mathcal{H} := \{h=f\circ g: f\in \mathcal{F}, g\in\mathcal{G}\}$.
QUESTION: Is the metric space $(\mathcal{H}, ||\cdot||_\infty)$ complete (Banach)? If so is it compact?
I know Arzela-Ascoli implies subsets of $C_B(X)$ are compact if and only if they are complete sets of uniformly equicontinuous and bounded functions, but I'm struggling to show that $\mathcal{H}$ is complete.