Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent.
I suspect the following to be true.
Let $(x_n)_{n=0}^\infty$ be a sequence in $V$. Then $(x_n)_n$ is a Cauchy sequence w.r.t. $\Vert \cdot \Vert_1$ if and only if $(x_n)_n$ is a Cauchy sequence w.r.t. $\Vert \cdot \Vert_2$.
Is it true?
I'm looking for a good way to explain why "equivalence" of norms is the right way of "comparing" norms without mentioning anything from topology besides "convergence".