Prove S(f,x+y) ≤ S(f,x) + S(f,y) for bounded functions

[Carl140]

Homework Statement

Let f: [0,1] -> R be a bounded function.

Define S(f,x) = sup { |f(r) - f(s) | : r,s in [0,1] and |r-s| <= x}.

Prove that if x>0, y>0 then:

S(f,x+y) <= S(f,x) + S(f,y).

The Attempt at a Solution

I have no idea how to proceed, could you please help?

 

[Billy Bob]

The statement to be proved has the form

sup{g(r,s) : r,s \in A and "condition"} <= M.

For the proof, try this: Let r0 and s0 be in A such that "condition." We must show g(r0,s0) <= M.

This method may not work in general, but for this problem I think it does. That will get you started. Write up something and ask again.