(a) State and prove the first Borel-Cantelli lemma. State the second Borel-Cantelli lemma.

(b) Let $X_{1}, X_{2}, \ldots$ be a sequence of independent random variables that converges in probability to the limit $X$. Show that $X$ is almost surely constant.

A sequence $X_{1}, X_{2}, \ldots$ of random variables is said to be completely convergent to $X$ if

$$\sum_{n \in \mathbb{N}} \mathbb{P}\left(A_{n}(\epsilon)\right)<\infty \quad \text { for all } \epsilon>0, \quad \text { where } A_{n}(\epsilon)=\left\{\left|X_{n}-X\right|>\epsilon\right\}$$

(c) Show that complete convergence implies almost sure convergence.

(d) Show that, for sequences of independent random variables, almost sure convergence also implies complete convergence.

(e) Find a sequence of (dependent) random variables that converges almost surely but does not converge completely.