Abstract: Fix an integer b greater than or equal to two, and for each positive integer n, let s(n) denote the sum of the digits of n when expressed in base b. Let S(n) be the cumulative digital sum, obtained by summing s(k) over k=0,1,...,n-1. I will present some inequalities for the function S(n). These have applications in mathematics (approximate convexity of real functions) and computer science (merge problems). The proofs are totally elementary, requiring nothing more than induction and perhaps a bit of creativity to see how the induction hypothesis may be applied.