Averages and Means

(2015 Bay Area Mathematical Olympiad-8 Problem C) Which number is larger, $A$ or $B$, where $A = \frac{1}{2015}\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)$ and $B = \frac{1}{2016}\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2016}\right)$? Prove that your answer is correct.