Proof: Note that the proof of (iii) is trivial, as it follows directly from the definitions of continuity and almost sure convergence. Of the remaining two parts, we’ll prove part (i) only.

The basic strategy of this proof is Portmanteau (c \(\to\) a), by which I mean we will show that if \(h\) is any continuous bounded function, then \(\Ex h(\x_n) \to \Ex h(\x)\). Convergence in distribution then follows from the Portmanteau theorem.

Let \(h: \real^k \to \real\) be a continuous bounded function, let \(f(\x) = h(g(\x))\), and let \(C(g), C(f)\) denote the continuity sets of \(g\) and \(f\), respectively.

\[\begin{alignat*}{2} &C(g) \subset C(f) &\hspace{4em}& \href{composition.html}{\text{Continuity of compositions}}; h \textnormal{ is continuous} \\ &f \textnormal{ continuous almost everywhere} && g \textnormal{ continuous almost everywhere} \\ &f \textnormal{ bounded} && h \textnormal{ bounded} \\ &\Ex h(g(\x_n)) = \Ex f(\x_n) \\ &\phantom{\Ex h(g(\x_n))} \to \Ex f(\x) && \href{portmanteau.html}{\text{Portmanteau}} (a \to d); \x_n \inD \x; f \textnormal{ bounded, continuous a.e.} \\ &\phantom{\Ex h(g(\x_n))} = \Ex h(g(\x)) \end{alignat*}\]