Completeness of a finite dimensional vector space under the metric induced by a norm

Question

I was studying linear algebra from "Linear algebra done right" by Sheldon Axler, and I found an exercise asking me to prove that any normed, finite dimensional vector space over the field of the real or complex numbers is a complete metric space with respect to the metric induced by the norm. I was not able to do any progress on this, so I searched online and I only found answers using the equivalence of 2 norms on a finite dimensional vector space, something that it's never been mentioned in the book so far, so I'm here asking if anyone could provide a proof that doesn't use this fact, thanks in advance

Answer 1

Exercise 29 in Section 6.A of Linear Algebra Done Right starts by defining a metric $d$ on the inner product space $V$ by $$d(u, v) = \|u-v\|.$$

In the statement of this exercise, $V$ is indeed an inner product space because 11 pages earlier there is a box stating "for the rest of this chapter, $V$ denotes an inner product space". Then norms throughout the chapter refer to the norm induced by the inner product. Thus to produce a solution to this exercise, it is not necessary to use the result that all norms on a finite-dimensional vector space are equivalent. A key tool to use in the solution is the result that every finite-dimensional inner product space has an orthonormal basis.