I'm trying to show that the integral from a to b of a function f is the same as the integral from a+c to b+c of the function f(x-c). In summary, I've tried to argue in the following way:
Say the integral is A. Let \epsilon>0 be given. Since f is integrable over [a,b], there is an arbitrary partition P_1 with mesh size less than \delta>0 such that the approximating Riemann sum S_1(P_1, f(x-c)) is within \epsilon of A. Then I defined a partition P_2 by adding c to each of the elements of P_1 and showed that the approximating sum S_2(P_2, f(x-c)) would be equal to S_1(P_1, f) and hence it wou…