How does one prove that the unit sphere $S^{n-1}$ in a finite dimensional real Banach space $V$ (with $\dim V = n$) is compact with respect to the topology induced by the norm? Because somehow I often see "proofs" that rely on
$$\text{$V$ is isomorphic and homeomorphic to $\mathbb{R}^n$} \\\Downarrow \\\text{$S^{n-1}$ is compact} \\ \Downarrow \\ \text{All norms on $V$ are equivalent} \\ \Downarrow \\ \text{All linear maps $L:V \to W$ are continuous} \\\Downarrow \\ \text{$V$ is isomorphic and homeomorphic to $\mathbb{R}^n$}$$ where $W$ is another finite dimensional real Banach space. This appears to be a vicious cycle.
Question:
What is a proof for one of these statements that does not rely on one of the previous statements?
One usually uses Heine-Borel theorem to show that the closed unit ball of $\mathbb R^n$ is compact. The arguments then go as follows: $$ {S^{n-1}\subset \mathbb R^n} \text{ is compact}\\ \Downarrow\\ \text{ all norms on }\mathbb R^n\text{ are equivalent}\\ \Downarrow\\ V \text{ is isomorph to } \mathbb R^n $$ This provides an entry into the circle.