Let be continuous and periodic with period For any real numbers evaluate
A continuous function is integrable, and the integral over a bounded interval of an integrable periodic function is bounded. Hence, using the substitution
since the integral is bounded
Something tells me I may have done something wrong, because, well, surely it can’t be that simple …
Let be the mean value of f over one period. Then the interval [na,nb] consists of subintervals of length T (not counting odd bits at the ends, which we can dispose of with epsilons). So . That's my candidate for the limit.