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Riemann integrals and step functions

  1. Apr 16, 2012 #1
    Prove the following:

    If f is Riemann integrable on an interval [a,b], show that ∀ε>0, there are a pair of step functions
    L(x)≤f(x)≤U(x)
    s.t.
    ∫_a^b▒(U(x)-L(x))dx<ε

    My proof:

    Since f is Riemann integrable on [a,b] then, by Theorem 8.16, ∀ε>0, there is at least one partition π of the interval [a,b] s.t.
    ∑_(k=1)^n▒〖ωf([x_(k-1),x_k ])(x_k-x_(k-1))〗<ε
    Let L(x)=inf⁡〖f(x)〗 ∀x∈[x_(k-1),x_k] and let U(x)=sup⁡〖f(x)〗 ∀x∈[x_(k-1),x_k]. Note that
    ωf([x_(k-1),x_k ])(x_k-x_(k-1) )=U(x)-L(x)
    so

    ∑_(k=1)^n▒〖(U(x)-L(x))(x_k-x_(k-1))〗<ε
    |∑_(k=1)^n▒〖(U(x)-L(x))(x_k-x_(k-1))〗-0|<ε
    Hence,
    ∫_a^b▒(U(x)-L(x))dx<ε
    Which is precisely the statement needed to be proven. ∎
    I feel like my proof makes sense, but I would like to get some feedback to see if anyone sees any flaws in my logic. Note that I essentially explained the details of Theorem 8.16 in my proof. Also, ignore the ▒'s, as they were created when copying and pasting from Word.
     
  2. jcsd
  3. Apr 16, 2012 #2
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