"Let (r_n) be any list of rational numbers in [0,1]. Let (a_n) be a sequence such that 1>a_1>a_2>...>a_n>... converging to 0. Let f be defined such that(adsbygoogle = window.adsbygoogle || []).push({});

f=0 if x is irrational

=a_n if x is equal to r_x

Prove that f is Riemann integrable."

We are doing integrals from Darboux' approach, so no tagged partitions or whatnot. I have to somehow show that the upper sum L(f,P) and lower sum L(f,P), for some partition P, differ by less than a given e>0.

L(f,P) = 0 for any partition, so |U(f,P) - L(f,P)|=U(f,P). I have no idea how to proceed, because if p is a partition of n equal parts, [tex]U(f,P) \leq \sum_{i=0}^{n}\frac{1}{n}[/tex] which diverges. I am unable to refine the inequality.

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# Proving that a function is Riemann integrable

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