Sequential Characterization of Integrability

A bounded function is Riemann integrable iff a sequence of partitions, such that:

In that case:

Proof Hint

Cauchy criteria and squeeze theorem is used for both side proof.

For :

• Consider the limit definition.
• Prove is Riemann integrable on by Cauchy criteria.
• Use squeeze theorem for to prove limit of upper sum
• Prove limit of lower sum using the limit of upper sum

For : Consider the below, where .

Theorem

Suppose is Riemann integrable on and . Then :

where .

Proof Hint