(a) State and prove Fatou's lemma. [You may use the monotone convergence theorem without proof, provided it is clearly stated.]

(b) Show that the inequality in Fatou's lemma can be strict.

(c) Let $\left(X_{n}: n \in \mathbb{N}\right)$ and $X$ be non-negative random variables such that $X_{n} \rightarrow X$ almost surely as $n \rightarrow \infty$. Must we have $\mathbb{E} X \leqslant \sup _{n} \mathbb{E} X_{n}$ ?