Theorem. If is Riemann integrable then its Riemann sums satisfy the Cauchy Criterion for the existence of the integral.

Cauchy Criterion: For every , there exists such that if is a partition of , with and and are Riemann sums of on , then .

Proof. Suppose is Riemann integrable on . Then for all , there exists such that whenever isaRiemann sum for corresponding to a partition of . Thus and . Hence .

Correct?

Also could we replace and with and . And so the inequality would then be: ? Because these are not necessarily Riemann sums.