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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:


every convergent sequence in $(X,d_1)$ is also convergent in $(X,d_2)$.


every Cauchy sequence in $(X,d_1)$ is also Cauchy in $(X,d_2)$.


every open set in $(X,d_1)$ is also an open set in $(X,d_2)$.


every open ball in $(X,d_1)$ is also an open ball in $(X,d_2)$.

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