Let f S rightarrow Ropf where S SubsetEqual Ropf Prove that

Let f: S rightarrow Ropf, where S SubsetEqual Ropf. Prove that if f is increasing on S or if f is decreasing on S, then f is an injection.

Solution

a)

Case 1: f is increasing (It must be strictly increasing)

Let is not be an injection

So there exist x,y so that, we can assume x<y without loss of generality

so that: f(x)=f(y)

But, f is increasing so f(x)<f(y)

So a contradiction

Hence, f is one to one

Case 2 f is decreasing

Hence, g=-f is increasing

And from Case 1. g is an injection and hence f is an injection