This seems like a pretty obvious statement to me but when I was going to use it in a proof I realize I need to prove it. Thanks!
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Does this (plus induction on $n$) answer your question? The product of two Riemann integrable functions is integrable – HallaSurvivor Jun 19 '22 at 03:05
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If $g$ is continuous and $f$ is Riemann integrable, then $g \circ f$ is Riemann integrable. This isn't obvious, but the proof is not too difficult. – Mason Jun 19 '22 at 15:12