Moderate# What If One Metric Dominates Another Metric?

TOPO-HXJ5KK

Let $X$ be a set and $d_1:X \times X \to (0,+\infty)$ and $d_2:X \times X \to (0,+\infty)$ be two metrics on $X$.

Suppose that there exists a constant $c\in (0,+\infty)$ such that

$$ d_1(x,y) \leq c d_2(x,y), \mbox{ for all }x,y \in X.$$

Then it is necessarily **TRUE** that: