Lemma 7.1.10: Riemann Lemma
Suppose f is a bounded function defined on the closed,
bounded interval [a, b]. Then f is
Riemann integrable if and only if for every
> 0 there exists
at least one partition P such that
| U(f,P) - L(f,P) | <
Proof:
One direction is simple: If f is Riemann integrable, then I^{*}(f) = I_{*}(f) = L. By the properties of sup and inf we know:- There exists a partition P such that L = I^{*}(f) > U(f, P) - / 2
- there exists a partition Q such that L = I_{*}(f) < L(f, Q) + / 2
- U(f,P) U(f,P')
- L(f,Q) L(f,P')
- L > U(f,P) - / 2 U(f,P') - / 2
- L < L(f,Q) + / 2 L(f,P') + / 2
0 > U(f,P') - L(f,P') -or equivalently:
> U(f,P') - L(f,P') = | U(f, P') - L(f, P')|Therefore we found a particular partition (namely P') such that
| U(f, P') - L(f, P')| <for any given .
The other direction is a little bit harder: Assume that for every > 0 we can find one partition P such that
| U(f, P) - L(f, P)| <We then need to show that I^{*}(f) - I_{*}(f)| <
We will do that later.