sequences and series - Inequality for finite harmonic sum

For a positive integer $n$ let
$$A(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}+\dots +\frac{1}{2^n - 1}$$
Then prove that $A(200) > 100 > A(100)$.

I tried some concepts like AM>GM>HM and some algebraic methods for reducing the series but was unable to solve it.
Please help me to solve this.