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Subdivisions/Refinement Proof

  1. Feb 27, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose f is a function bounded on [a,b], A=GLB(S u,f), and B=LUB(S W,f).


    2. Relevant equations

    1. For each e>0, there is a subdivision D={Xi}of [a,b] such that |U f,D - W f,D|<e
    2. There is a number Q such that if e>0, then there is a subdivision D of [a,b] such that if K={Yi}is a refinement of D, then |U f,K - Q|<e and |W f,K - Q|<e.

    U=upper sums
    W=lower sums

    3. The attempt at a solution

    I'm supposed to show that 1 implies 2 and 2 implies 1. Trying to do 1 implies 2 confuses the hell out of me, but if I do 2 implies 1 isnt that just doing a triangle inequality and account for the number Q?
     
  2. jcsd
  3. Feb 28, 2013 #2
    Your notation is not very clear here. What is S? Is this for the Riemann-stieltjes integral, and S is the monotonically increasing function of integration? In your question the S doesn't seem to come into play, so it's not too big a deal..

    For 1 implies 2, do you have the theorem that if P is a refinement of D then L(D,f)<L(P,f)<U(P,f)<U(D,f)? (where < is supposed to be less than or equal too but I'm lazy on my ipad).
     
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