1. The problem statement, all variables and given/known data I'm trying to show that a distance preserving map is 1:1 and onto. The 1:1 part was easy, but I'm stuck on proving it's onto... 2. Relevant equations X is compact T(X)[itex]\subseteq[/itex]X THere's a hint saying to consider a point y in X\T(X) and consider the minimum distance between y and x[itex]\in[/itex] T(X) (the infinum of d(x,y) for all x[itex]\in[/itex] T(X) where d is an undefined metric, and call it deltA) Then it says to consider the sequence yn=Tn(y) 3. The attempt at a solution Because X is compact, and yn is a subset of X, it must be bounded (I think?) and have a convergent subsequence, and inf(d(T^n(y),T^n(x))=delta for all n (and x_n=T^n(x) will also have a convergent subsequence). Show that the limits of y_n and x_n provide a contradiction?