Let $X$ be a metric space with the metric $d: X \times X \rightarrow \mathbb{R}$.

(i) Show that if $X$ is compact as a topological space, then $X$ is complete.

(ii) Show that the completeness of $X$ is not a topological property, i.e. give an example of two metrics $d, d^{\prime}$ on a set $X$, such that the associated topologies are the same, but $(X, d)$ is complete and $\left(X, d^{\prime}\right)$ is not.