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Math Help - Riemann Integral Proof

  1. #1
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    Riemann Integral Proof

    I am having a bit of trouble with this proof and any insight would be appreciated.

    Suppose f is a nonnegative Riemann integrable function on [a,b] satisfying f(r) = 0 for all r in \textbf{Q} \cap  [a,b]

    Prove that  \int_a^b f = 0
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  2. #2
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    I will try to help even though the definitions you are using may be different but equivalent.
    Recall that between any two numbers there is a rational number.
    Thus, given any division of [a,b] then on any subinterval the lower bound is zero.
    Thus given \varepsilon >0 there is a division of [a,b] D such that \left| {\overline {\int_D f }  - \underline {\int_D f } } \right| = \left| {\overline {\int_D f } } \right| < \varepsilon
    That implies that the integral is zero.
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