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The goal is to show that the product of two Riemann integrable functions is integrable.

First step is to use the identity $f\cdot g = \frac{1}{4} \left[(f+g)^2 - (f-g)^2\right]$ so that we only need to consider squares of functions.

The second step is to reduce to positive valued functions because $f(x)^2=\left|f(x)\right|^2$.

The third step is to use that if $0 \leq f(x) \leq M$ on $\left[a,b\right]$, $$f^2(x) - f^2(y) \leq 2M \left(\,f(x)-f(y)\right)$$

How should I go about implementing the above steps?

  • 11
    This is homework, right? (Can you capitalize your sentences, by the way? It makes for much more pleasant reading!)2010-09-29
  • 0
    You seem to have outlined a proof sketch in your question. Namely, you have outline how to reduce to the case of squares, and then how to show that the difference of the square of the values of $f$ at nearby points is bounded by a scalar times the difference of the values of $f$. The next step will be to look at the Riemann sums for $f^2$, and control them in terms of the Riemann sums for $f$, using the bound you have proved.2010-09-30

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