Finite dimensional normed spaces

I find this problem interesting since it gives another characterization of finite dim. normed spaces.

It is a standard result that every finite dim. normed space (over $\displaystyle \mathbb{F} = \mathbb{R} , \mathbb{C}$ ) is complete. Prove the converse:

If $\displaystyle V$ is a vector space over $\displaystyle \mathbb{F}$ such that for every norm $\displaystyle \Vert \cdot \Vert : V \rightarrow \mathbb{R}$ we have that $\displaystyle (V,\Vert \cdot \Vert )$ is complete then $\displaystyle \dim (V) < \infty$

I include a hint and the solution for those that want them.

Hint

Solution

One thing that also came to mind, but haven't given much thought (not posting it as a question but as recreation) if $\displaystyle V$ is such that every two norms are comparable, does it follow $\displaystyle \dim (V) < \infty$ ? (The weaker statement using that two norms are equivalent follows directly from the posed problem).