In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in $\mathbb R^n$. This exercise succeeds another exercise where various equivalent formulations of equivalent norms have been given, e.g. that two norms $\|\cdot\|_a,\|\cdot\|_b$ are equivalent iff the identity map from $(\mathbb R^n,\|\cdot\|_a)$ to $(\mathbb R^n,\|\cdot\|_b)$ is bicontinuous.
Now, in showing that all norms in $\mathbb R^n$ are equivalent, the authors show a given norm $\|\cdot\|$ is equivalent to the $1$-norm (and then by transitivity, we have equivalence for all norms). I have already managed to understand that the identity is continuous from $(\mathbb R^n,\|\cdot\|_1)$ to $(\mathbb R^n,\|\cdot\|)$. To show that the inverse of the identity map is continuous, the authors claim that the unit sphere in the $1$-norm is compact. I'm getting hung up on this statement, since I don't know how to go about this without using that the norms are equivalent already. How would one show the unit sphere in the $1$-norm is compact?
I know of Heine-Borel, but I'm not sure how and if it applies here. I've struggled with this statement and any help would be very appreciated. If you think there's another answer already that answers this in a concise and comprehensive way, please comment.