# Monotone function with predescribed discontinuities

Let I(x) (the unit jump function) be defined piecewise by I(x)=0 iff x<0 and I(x)=1 iff x>=0.

Let {x_n} be a countable subset of R and {c_n} a sequence of positive real numbers satisfying \sum{c_n}=1. Let f:R \to R be defined by f(x)=\sum{c_n * I(x - x_n)}.

Prove that \lim_{x\to-\infty}{f(x)}=0 and \lim_{x\to\infty}{f(x)}=1.

I can prove this when {x_n} is a countable subset of (a,b) where a,b \in R. It seems as though the proof here should be very similar, but for some reason I just can't finish it..

EnumaElish