There is an easy way to prove two numbers are equal. Typically in the reals there are three possibilities: a > b, a < b, a = b. If you eliminate a > b and a < b then you are left to conclude a = b. And this is exactly what is done in Apostol's Calculus Vol 1 (IMHO the greatest calc book ever written) chapter 1 when he proves that the area under n^2 is EXACTLY (n^3)/3, with no "calculus". You would be shocked how far into calculus the author gets with just that theorem. Can't recommend that book enough.