Which number is larger, A or B, where
A=\frac{1}{2015}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2015}\right) \quad \text { and } \quad B=\frac{1}{2016}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2016}\right) ?
Prove that your answer is correct.