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Upper & Lower Partitions

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose f:[a,b] [tex]\rightarrow[/tex] [tex]\Re[/tex] is bounded and that the sequences {[tex]U_{P_{n}}[/tex](f)}, {[tex]L_{P_{n}}[/tex](f)} are covergent and have the same limit L. Prove that f is integrable on [a,b].

    2. Relevant equations

    [tex]U_{P_{n}}[/tex](f) is the upper sum of f relative to P, and [tex]L_{P_{n}}[/tex](f) is the lower sum of f relative to P, where in both cases P is the partition of [a,b].

    Also a hint was to examine the sequence {[tex]a_{n}[/tex]} where [tex]a_{n}[/tex] = [tex]U_{P_{n}}[/tex](f) - [tex]L_{P_{n}}[/tex](f)

    3. The attempt at a solution

    Didn't figure out the full proof, only parts:

    I started with having P = {[tex]x_{0}[/tex],..., [tex]x_{n}[/tex]}, where [tex]x_{0}[/tex] = a, and [tex]x_{n}[/tex] = b. And given that they are both convergent and have the same limit, using the hint I can show that they both converge to [tex]a_{n}[/tex], but I got stuck after that.
     
  2. jcsd
  3. Jan 27, 2009 #2
    It's integrable if a single value exists between the two sequences as n tends to infinity, right? I'd look at this as a nested sequence of intervals, where the length of the nth interval as n approaches infinity is 0 (because the two sequences have the same limit), and from there it should be easy.
     
    Last edited: Jan 27, 2009
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