Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, for each $n\in\mathbb{N}$ let $X_{n}$ be a random variable on a probability space $(\Omega_{n},\mathcal{F}_{n},P_{n})$, with $\mathbb{E}_{n}$ the expectation there, and let $X$ be a random variable on $(\Omega,\mathcal{F},P)$.

(i) Smooth test functions. $X_{n}\xrightarrow{d}X$ if and only if

$$\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)\qquad(n\to\infty)$$

for every $f:\mathbb{R}\to\mathbb{R}$ such that $f$ and its successive derivatives $f'$ and $f''$ are differentiable, $f'''$ is continuous, and there is a real $M$ with $|f|,|f'|,|f''|,|f'''|\le M$ on $\mathbb{R}$.