Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that
$d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$.
- Show that if $E \subset X$ is an open set in the metric space $(X, d_1)$, then $E$ is also an open set in the metric space $(X, d_2)$.
- Show that if $(X, d_2)$ is a compact metric space, then $(X, d_1)$ is also a compact metric space.