Why do Riemann sums converge?

Question

I know from calculus that Riemann Integrals are Riemann sums. We use such integrals to calculate areas under curves. But why does adding an infinite amount of small terms give a finite result for non divergent curves? I know the obvious answer would be to just look at the graph of the function and we can see there is a finite area. But I keep thinking of the Harmonic series and how adding an infinite amount of smaller and smaller terms has as a result infinity. How can we be sure the small terms if the Riemann sum also don't add up to infinity?