With the usual metric d_1, where , V is incomplete. So the problem is to find another metric d_2, under which V is complete. Use the fact that that the interval (0,1) is homeomorphic to the whole real line. The homeomorphism can be used to transport the usual metric on across to the unit interval, thereby giving it a complete metric.

Specifically, if 0,1)\to \mathbb{R}" alt="f0,1)\to \mathbb{R}" /> is a homeomorphism, then you can take .