I actually have two questions I am having trouble with. 1. The problem statement, all variables and given/known data What is the completion of a discrete metric space X? 2. Relevant equations d(x,x) = 0 d(x,y) = 1 if x does not = y I don't really understand how to complete a metric space that is incomplete. I just know that every Cauchy sequence in X would have to converge to something in X itself, but I'm not sure how to manipulate it to ensure that this happens. AND 1. The problem statement, all variables and given/known data If X1 and X2 are isometric and X1 is complete, show that X2 is complete. I know that since they are isometric, there is a mapping T such that d2(Tx,Ty) = d1(x,y). Other than that, I'm not sure how to prove it. It just kind of seems intuitive. Any suggestions would be GREATLY appreciated. Thank you SO much.