(i) Let $X$ be the set of all infinite sequences $\left(\epsilon_{1}, \epsilon_{2}, \ldots\right)$ such that $\epsilon_{i} \in\{0,1\}$ for all $i$. Let $\tau$ be the collection of all subsets $Y \subset X$ such that, for every $\left(\epsilon_{1}, \epsilon_{2}, \ldots\right) \in Y$ there exists $n$ such that $\left(\eta_{1}, \eta_{2}, \ldots\right) \in Y$ whenever $\eta_{1}=\epsilon_{1}, \eta_{2}=\epsilon_{2}, \ldots, \eta_{n}=\epsilon_{n}$. Prove that $\tau$ is a topology on $X$.

(ii) Let a distance $d$ be defined on $X$ by

$$d\left(\left(\epsilon_{1}, \epsilon_{2}, \ldots\right),\left(\eta_{1}, \eta_{2}, \ldots\right)\right)=\sum_{n=1}^{\infty} 2^{-n}\left|\epsilon_{n}-\eta_{n}\right|$$

Prove that $d$ is a metric and that the topology arising from $d$ is the same as $\tau$.