Here's a proof of the equivalence of norms in finite dimension that does not use compactness arguments. It breaks down into three steps.
We set some notation first: given a normed finite dimensional real vector space $E$, write $E'$ for its algebraic dual space (set of all linear forms) and $E^{*}$ for its topological dual space (set of all continuous linear forms).
- All linear forms on $E$ are continuous, i.e $E'=E^{*}$, whatever the norm $\|.\|$ on $E$.
The Hahn-Banach theorem states that every continuous form on a subspace of $E$ can be extended to all of $E$. In particular, given
$x\in E$, there is a $\varphi:E\rightarrow
\mathbb{R}$ for which $\varphi(x)\neq 0$, since there are non trivial forms $\mathbb{R}x\rightarrow \mathbb{R}$ and these can be extended. As a consequence, the map
\begin{align*}
E &\rightarrow (E^{*})' \\
x &\mapsto (\phi\rightarrow \phi(x))
\end{align*}
is injective. We also have isomorphisms $E'\simeq E$ and $E^{*}\simeq (E^{*})'$. Composing the three maps we have an injection $E' \rightarrow E^{*}$ so that $\dim E' \leqslant \dim E^{*}$. On the other hand since $E^{*}\subset E'$ we must have $\dim E^{*}=\dim E'$ and $E^{*}=E'$ as desired.
- All linear maps between finite dimensional normed spaces are continuous.
Let $A:F\rightarrow E$ be such a map and $e_{i},i\in I$ a basis for $E$. Let $(e_{i}^{*})$ be the dual basis of $(e_{i})$ and $\tilde{e_{i}}$ the linear map $\mathbb{R}\rightarrow E$ that maps $1$ to $e_{i}$. Then $Id_{E}=\sum_{i\in I}\tilde{e_{i}}e_{i}^{*}$. In particular, $A=\sum \tilde{e_{i}}e_{i}^{*}A$. The the $\tilde{e_{i}}$ are continuous and so are the $e_{i}^{*}A$, since they are linear forms. Hence $A$ is continuous as a sum of composites of continuous maps.
- All norms are equivalent.
Given two norms $\|.\|_{1}, \|.\|_{2}$ on $E$, the identity, being linear, defines an homeomorphism between $E,\|.\|_{1}$ and $E,\|.\|_{2}$. It is easy to check that this means that the norms are equivalent.
This proof is really a way of saying that the topology induced by a norm on a finite-dimensional vector space is the same as the topology defined by open half-spaces; in particular, all norms define the same topology and all norms are equivalent. There are other ways to prove that using the Hahn-Banach theorem.