Convergence in Distribution of Real Random Variables through Distribution Functions

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 $(\Omega_{n},\mathcal{F}_{n},P_{n})$, and let $X$ be a random variable on $(\Omega,\mathcal{F},P)$.

(i) Convergence of the distribution functions. If $X_{n}\xrightarrow{d}X$, then

$$F_{X_{n}}(t)\to F_{X}(t)\qquad(n\to\infty)$$

for every $t\in\mathbb{R}$ at which $F_{X}$ is continuous.

(ii) The converse. If $F_{X_{n}}(t)\to F_{X}(t)$ for every $t\in\mathbb{R}$ at which $F_{X}$ is continuous, then $X_{n}\xrightarrow{d}X$.