# Interactive Real Analysis

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## 7.1. Riemann Integral

### 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
Take the partition P' that is the common refinement of P and Q. Then we know that:
• U(f,P) U(f,P')
• L(f,Q) L(f,P')
Taking this together we have:
• L > U(f,P) - / 2 U(f,P') - / 2
• L < L(f,Q) + / 2 L(f,P') + / 2
Multiplying the second inequality by -1 and adding it to the first gives:
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.

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