Property of topologically equivalent metrics

Question

Let $X$ be a compact topological space, and $d_1,d_2$ two metrics that induce the topology on $X$. Is it necessarily true that for every $\epsilon > 0$ there exists a $\delta > 0$ such that: $d_1(x,y) < \delta \rightarrow d_2(x,y) < \epsilon$?

Answer 1

Hint 1: Consider the identity map $i : X \to X$ as a map between the metric spaces $(X, d_1)$ and $(X, d_2)$. Then $i$ is a homemorphism, i.e. $i, i^{-1}$ are continuous.

Hint 2: A continuous function on a compact metric space is uniformly continuous.

If you unwind the proof of Hint 2, you should be able to come up with a direct proof of your claim.