Let $\|\cdot\|$ be any norm on $\mathbb R^n$ prove that a sequence $x \in \mathbb R^n$ is a Cauchy sequence under $\|\cdot\|_2$ if and only if it is a Cauchy sequence under any $\|\cdot\|$.
I want to use the system from the post and say for a Cauchy sequence $|x_m-x_n|<\varepsilon$ and therefor $\|x_m-x_n\|_1<\varepsilon$, and then use $c\|x_m-x_n\|_2\leq \|x_m-x_n\|_1\leq d\|x_m-x_n\|_2$.
Can i say that if $|x_m-x_n|<\varepsilon$ (from wiki) then the norm is also smaller then $\varepsilon$ that is: $\|x_m-x_n\|<\varepsilon$.
Can I apply the same system as used in the post in this case? if not what will be a good way to prove this?