(i) State and prove Kolmogorov's zero-one law.

(ii) Let $(E, \mathcal{E}, \mu)$ be a finite measure space and suppose that $\left(B_{n}\right)_{n \geqslant 1}$ is a sequence of events such that $B_{n+1} \subset B_{n}$ for all $n \geqslant 1$. Show carefully that $\mu\left(B_{n}\right) \rightarrow \mu(B)$, where $B=\cap_{n=1}^{\infty} B_{n}$.

(iii) Let $\left(X_{i}\right)_{i \geqslant 1}$ be a sequence of independent and identically distributed random variables such that $\mathbb{E}\left(X_{1}^{2}\right)=\sigma^{2}<\infty$ and $\mathbb{E}\left(X_{1}\right)=0$. Let $K>0$ and consider the event $A_{n}$ defined by

$$A_{n}=\left\{\frac{S_{n}}{\sqrt{n}} \geqslant K\right\}, \quad \text { where } \quad S_{n}=\sum_{i=1}^{n} X_{i}$$

Prove that there exists $c>0$ such that for all $n$ large enough, $\mathbb{P}\left(A_{n}\right) \geqslant c$. Any result used in the proof must be stated clearly.

(iv) Prove using the results above that $A_{n}$ occurs infinitely often, almost surely. Deduce that

$$\limsup _{n \rightarrow \infty} \frac{S_{n}}{\sqrt{n}}=\infty$$

almost surely.