Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 2 / d

Let $g:M^{\prime }\to \mathbb{R}$ be bounded and continuous. The composition $h=g\circ f$ is bounded and measurable, and every discontinuity point of $h$ is a discontinuity point of $f$. ThusThe Portmanteau theorem includes the null-discontinuity criterion: if $X_n\stackrel{d}{\to }X$ and a bounded measurable function is continuous at $X$ almost surely, then its expectations converge. HenceSince this holds for every bounded continuous $g$, it proves the continuous mapping theorem conclusion $f(X_n)\stackrel{d}{\to }f(X)$.