Let $X$ be a compact metric space with two metrics $d,d'$ inducing the same topology on $X$.
Show or find a counterexample to:
There exists a sequence $(\delta_n)_n$ such that $\delta_n\rightarrow 0$ such that for every $\epsilon>0$ if $d(x,y)<\epsilon$ then $d'(x,y)<\delta_n$ for any large $n \in \Bbb N$.
Thoughts:
(1). Since $d,d'$ induce the same topology, it follows that for every $\epsilon>0$ there exists $\delta>0$ such that if $d(x,y)<\delta$ then $d'(x,y)<\epsilon$ .
(2). Assume there exists $\epsilon>0$ such that for each $n$, there exists $x_n,y_n\in X$ such that if $d(x_n,y_n)<\epsilon$ then $d'(x_n,y_n)\geq \frac{1}{n}$. Since $X$ is compact, assume WLOG $x_n\rightarrow x$ and $y_n\rightarrow y$. Then, $d(x,y)\leq \epsilon$