(i) What does it mean to say that a sequence of random variables $\left(X_{n}\right)$ converges in probability to $X$ ? What does it mean to say that the sequence $\left(X_{n}\right)$ converges in distribution to $X$ ? Prove that if $X_{n} \rightarrow X$ in probability, then $X_{n} \rightarrow X$ in distribution.

(ii) What does it mean to say that a sequence of random variables $\left(X_{n}\right)$ is uniformly integrable? Show that, if $\left(X_{n}\right)$ is uniformly integrable and $X_{n} \rightarrow X$ in distribution, then $\mathbb{E}\left(X_{n}\right) \rightarrow \mathbb{E}(X)$.

[Standard results from the course may be used without proof if clearly stated.]