How to show that the harmonic function $H(n) = 1 + \frac{1}{2} + · · · + \frac{1}{n} = O(\log n)$ using simple inequalities on fractions? How can I prove that $H(n) = 1 + \frac{1}{2} + · · · + \frac{1}{n} = O(\log n)$ using simple inequalities on fractions?

Hint. Prove by induction that
$$1+\frac{n}{2}\leq H_{2^n}\leq 1+n$$
by using the fact that
$$\frac{1}{2}=\frac{2^n}{2^n+2^n}\leq H_{2^{n+1}}-H_{2^n}=\frac{1}{2^n+1}+\frac{1}{2^n+2}+\cdots +\frac{1}{2^{n}+2^n}\leq \frac{2^n}{2^n+1}<1.$$
