It is not necessary to look at upper and lower sums. Given any specific partition, S, of [a, b], and any set of x values, , consisting of one point in each subset, call that sum A_S. Because f is integrable, the limit, using any partition and any choice of x values, exist and is equal to the integral of f over [a,b]. Using the same partition and x values for g, gives (1/2)A_S, which clearly also converges to 1/2 the integral of f over [a,b].