Definition. A norm $\|\cdot\|$ in a vector space $X$ is said to be equivalent to a norm $\|\cdot\|_0$ on $X$ if there are positive numbers $a$ and $b$ such that for all $x \in X$ we have
$$ a\| x \|_0 \leq \|x\| \leq b\|x\|_0 $$
My question. If two norms $\|\cdot\|$ and $\|\cdot\|_{0}$ on a vector space $X$ are equivalent, then $\|x_{n} - x\| \rightarrow 0$ if and only if $\|x_n - x\|_0 \rightarrow 0$.
I know that two equivalents norms induce same the topology. How can I use it to prove the sentence.
Ref: (Kreyszig) Introductory Functional Analysis with Applications.