This note wasn't a planned one, but yesterday I tried to solve a very beautiful problem proposed to me by Gabriel Stuart Romon. The problem asked to find the following limit
The detailed solution to this problem, written by Gabriel may be found here. However, I want to discuss another problem, which I found on math.stackexchange. The key idea of it helped me to elaborate the solution to Gabriel's problem. I hope you'll find it beautiful and interesting.
Problem. If is a continuous periodic function with period 1 and is a continuous function, prove that
Solution. First of all we'll break our single integral into the sum of integrals over the shorter intervals.
By applying mean-value theorem we'll transform our sum into something similar to a Riemann sum and we'll use the fact that is periodic in our favor.
Because every , we have that the limit of our sum is , so