Given a normed, finite-dimensional vector space $V$, where we define the metric $d(x,y)=\|x-y\|$. Show that this space, under this metric, is complete - that is, every Cauchy Sequence converges.
I'm looking for, preferably, a linear algebra-based approach if possible. I've already tried writing this in terms of the basis and studying the coefficients. I also looked at the sequence $\| x_n\|$ but I haven't made much connections.
