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Riemann integrable sequences of functions

  1. May 17, 2010 #1
    Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx

    I am really lost, so any kind of help would be greatly appreciated.
    The part that confuses me the most is "for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k"

    It seems that the sequence of functions are integrable on [0,1] but there is a problem at 0 because of sup (1/k<x ≤1). This means that we need to split the integral into to parts, one from 0 to 1/k and the other one from 1/k to 1. In the end we would see that having a point on the interval that is problematic does not make the function not Riemann integrable.
    Am I correct?
    Once I split the integral should I just integrate both of them using the definition (Riemann integral)?
    Last edited: May 18, 2010
  2. jcsd
  3. May 18, 2010 #2
    When you say 0 is problematic, I took it to mean that nothing in the problem restricts it from being any value at all inside the codomain. Yes, that does not mean the function is not Riemann integrable. Individual points are in a sense not seen by integration.

    Remember that you are trying to prove that f is Riemann integrable, so you cannot split it up and integrate it. What you should do is recall what property f must satisfy in order for it to be Riemann integrable, and see if you can prove that property (now by splitting up the domain properly based off what you suggested).
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