Where can I learn about the formal proof for the Riemann Sum / Integral?

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In my calculus class, we started with some real analysis but the professor told us that this year he won't be going over the formal proof for the Riemann sum/integral. I looked it up and only found different proofs for Riemann properties as well as other concepts. Does anyone know what he may have been talking about? What are some resources I can use to understand it?

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Everything you are interested about, you'll find in Walter Rudin's "Principles of Mathematical Analysis".