Smooth Test Function Criterion for Convergence in Distribution

Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be random variables, not necessarily on a common probability space, and let $\mathbb{R}$ denote the real numbers. Call a function $f:\mathbb{R}\to\mathbb{R}$ an admissible test function if $f$ is bounded, $f$ is a $C^3$ map on $\mathbb{R}=\mathbb{R}^1$, and its first, second, and third derivatives are bounded. Then the following hold.

1. For every admissible test function $f$ and every random variable $Y$, the composition $f\circ Y$ is a random variable with finite expectation; in particular $\mathbb{E}[f(X_m)]$ and $\mathbb{E}[f(X)]$ are defined real numbers.

2. If

for every admissible test function $f$, then $X_m\to X$ in distribution, in the sense of Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution.