I am having a bit of trouble with this proof and any insight would be appreciated.
Suppose f is a nonnegative Riemann integrable function on [a,b] satisfying f(r) = 0 for all r in
I will try to help even though the definitions you are using may be different but equivalent.
Recall that between any two numbers there is a rational number.
Thus, given any division of [a,b] then on any subinterval the lower bound is zero.
Thus given there is a division of [a,b] such that
That implies that the integral is zero.