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Homework Help: Help real analysis

  1. May 6, 2009 #1
    1. The problem statement, all variables and given/known data

    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).

    3. The attempt at a solution

    I have no idea how to proceed, could you please help?
  2. jcsd
  3. May 6, 2009 #2
    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.
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