Norms Equivalence over $\mathbb R^n$

Question

Let $\|\cdot\|$ be any norm on $\mathbb R^n$.

Prove that a sequence in $\mathbb R^n$ is Cauchy under the $\|\cdot\|_2$ norm if and only if the sequance is Cauchy under the $\|\cdot\|$.

Prove that a set in $\mathbb R^n$ is bounded under $\|\cdot\|_2$ if and only if the sequence is bounded under the $\|\cdot\|$

Is there a general way for these type of questions or are they different for each case?

Answer 1

In a finite dimensional real (or complex) vector space all norms are equivalent. That means (following your notation) there are $c,d\in \mathbb R$ such that $$c||x||\leq ||x||_2\leq d||x||$$

Now the properties you want to show follows from this inequality.