The reason Plato asked if you knew about Riemann sums is that how you prove such a basic property of integrals depends upon how you define integrals in the first place! And "Riemann sums" is a common method of defining integrals. You mention of "upper sums" indicates that's what you have learned, although "upper sums" is only a very small part of that.
Given that f(x) is integrable on [a, b], then for any partition of [a, b] (dividing it into smaller subintervals), choosing any in the interval, with width , the Riemann sum is and, as we take "finer and finer" partitions, meaning that the largest goes to 0 as we take more and more intervals in the partition, and that sum goes to some fixed value, I, whic is the integral, .
Now, if for all x in [a, b], in particular, for every partition and for every in a partition. That, in turn, means that for every partition and every i in a partition, and so for every partition. The integral, the limit of negative numbers can't be positive: .