Problem Statement: (a) State a definition of a Riemann Integral. (b) Use this definition to prove if are both bounded, Riemann Integrable, and defined on , then is Riemann Integrable on .
(a) A bounded function is Riemann Integrable on if for every there exists a such that for a partition of with it follows that .
Proof: Since are both bounded there exist such that for every and for every .
Let . Since both are Riemann Integrable on we know that there exist such that for any partition of with it follows that . Similarly for any partition with it follows that . Let and be a partition of such that . Then it follows that
For space reasons I am going to drop the subscript notation on the sup and inf.
Thus, is Riemann Integrable on .
Reflection: When I first approached this problem my first thought was to use the sup-inf definition of R.I. since we were told that the functions were both bounded. The part I got stuck on was how to relate to and . I figured there was some trick to it, something like adding and subtracting the same thing to break the pieces apart. Ultimately that’s what we did but we got there in a different way. We wrote down what we wanted to end up with and worked backwards to figure out what we needed to add and subtract. Now I feel like I could come up with this trick on the actual exam if I needed to.