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Ratpigeon
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Homework Statement
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...
Homework 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)
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?