Mathematics — Real Analysis · continuous-functions
Problem
A continuous f:X→Y between metric spaces (X,dX) and (Y,dY) is said to be proper if for every compact set K⊂Y, the set f−1(K) is compact. Suppose a continuous f:(0,1)→(0,1) is proper and {xn}n=1∞ is a sequence in (0,1) converging to 0. Show that {f(xn)}n=1∞ has no subsequence that converges in (0,1).