MathCS.org - Real Analysis: Lemma 7.1.10: Riemann Lemma

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) | <

Context

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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Lemma 7.1.10: Riemann Lemma

Proof: