To prove the limit is equal to the integration, we need the definition of Riemann Integral and the uniform continuity of f(x).

that is, it is suffice to prove that for any

, there exist N, such that if k > N, then the longest distance of two consecutive( the oder of number, not the index n) among the previous k points is less than

, Or in another word, the distance of any two consecutive points of the previous k points is less than

. Although I understand what is actually going on, I can't express by word, can't write down the details here!