Since is finite dimensional and is a norm it follows that is a Banach space.

If is a Cauchy sequence in , then for there exist such that for . A direct calculation shows that this implies

Taking small enough (the quotient in the last inequality goes to 0 as goes to 0), this shows that is a Cauchy sequence in . Hence there

such that tends to in . But this implies that tends to in , since

.

Thus the answer is yes if you show that is actually a metric.