Define $\displaystyle a_n = 1+\frac{1}{2}+\frac{1}{3}+ \cdots + \frac{1}{n} - \int^n_1 \frac{1}{t}dt$. Prove that $\displaystyle a=\lim_{n \rightarrow \infty} a_n$ exists and $\displaystyle 0<a<1$.

Attempt

My issue is the integral in the sequence. I don't know how to take the limit of the sequence with that in there. I think there is a way to break up the integral to take the limit of the sequence easier. I need help with this question. Thanks in advance.