Q: Prove that a bounded function is integrable on if and only if there exists a sequence of partitions satisfying

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Here some of my (incomplete) thoughts:

Let . Since there exists a sequence of partitions satisfying , there exists an such that, for any . Thus, integrable on .

For this direction I am not sure how to show there exists a sequence of partitions. Since we are assuming is integrable, I know that there is a partion of such that for any positive .