Finite-dimensional normed spaces are complete

Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$. Then $V$ is a Banach space.

Remarks

Proof: by equivalence of norms in finite dimension, the norm is equivalent to the sup norm of the coordinates in a basis. Equivalent norms have the same Cauchy sequences and the same convergent sequences. With the sup norm, completeness follows from the completeness of $\mathbb{R}$ in each coordinate. A consequence: each finite-dimensional subspace of a normed space is complete, and thus closed.

Explanations

L1 human

In finite dimensions, a Cauchy sequence is Cauchy in each coordinate, so each coordinate converges. This works for each norm, because all norms are equivalent there.