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I know we haven't even merged derivatives yet, but I think this is actually well within our reach. The hardest part, I think, is simply showing that the equivalence of Fin mn and Fin m x Fin n has the expected relationship where (k, l) maps to nk + l, which is probably the key step in showing the limit converges.
From here, I think it's actually not a large step to Theorem 15 on the 100 Theorems, the Fundamental Theorem of Integral Calculus.