$\| \|_1$and $\| \|_2$ are equivalent in a finite dimensional vector space Let $V$ be a finite dimensional vector space. Let $\| \|_1$and $\| \|_2$ be norms on $V$. Show that if $y \in V$ and $(x_1,x_2,..)$ is a sequence in $V$ then $\lim_{n \to \infty} \|x_n-y\|_1=0$ if and only if $\lim_{n \to \infty} \|x_n-y\|_2=0$.
I have shown that if $\lim_{n \to \infty} \|x_n-y\|_1=0$ then $\lim_{n \to \infty} \|x_n-y\|_2=0$ but I don't know how to show the other part.
 A: There's a bit of ambiguity regarding who $\Vert\cdot\Vert_1$ and $\Vert\cdot\Vert_2$ are, but if I understood well you want to show that any two norms on a finite dimensional vector space are equivalent. So let $dimV=n$ and let $\{e_1,\ldots,e_n\}$ be a basis of $V$. Then the basis defines an isomorphism of $V$ with $\mathbb{R}^n$, since any vector $v\in V$ can be written uniquely as $a_1e_1+\ldots+a_ne_n$ and so can be identified with $(a_1,\ldots,a_n)\in\mathbb{R}^n$. But then any norm $\Vert\cdot\Vert_V$ on $V$ induces a norm on $\mathbb{R}^n$, given by $\Vert (a_1,\ldots,a_n)\Vert_{\mathbb{R}^n}=\Vert v\Vert_V$. In particular, if you already know the result that states that any two norms on $\mathbb{R}^n$ are equivalent, you can deduce that it holds also on any finite dimensional vector space. If you don't know the result, in order to show that any two norms on $V$ are equivalent, it suffices to show that any norm is equivalent to the following one:
$$\Vert a_1e_1+\ldots+a_ne_n\Vert := \sum_{i=1}^n \vert a_i\vert$$
In order to do this, you need to use inequalities similar to the ones of @olsen5 comment.
