We say that two metrics d, d' on a space S are equivalent if each "dominates" the other in the following sense: there exist constants M, M'>0 such that
d'(x,y)<=M' d(x,y) and d(x,y)<=M d'(x,y) for all x,y in S.
If metrics d, d' are equivalent, prove that (S,d) is complete<==>(S,d') is a complete metric space.
The Attempt at a Solution
If (S,d) is complete then every Cauchy sequence in S converges to a limit in S. I want to go somewhere along the lines of saying that multiplying by a constant will not change its convergence.....am I going along the right lines here? I don't really know how else to go about this.