Theorem. Let and be real numbers with and . Let and be real-valued functions that are Riemann integrable on . Then the following hold:

1. If for all then .

2. If for all then .

3. If for all , then .

Outline of Proof.Basically we consider the values and can take, and look at and ? For instance, in #2 we look at (which is Riemann integrable) and show that ?