Let $\displaystyle H_n$ be the $\displaystyle n$th harmonic number. I started with induction and easily arrived at $\displaystyle H_n$ is not an integer so assume that $\displaystyle H_n=\frac{p}{q}$ where $\displaystyle \gcd(p,q)=1$. Then $\displaystyle H_{n+1}=\frac{p}{q}+\frac{1}{n+1}$.

But I realize now that this is a dead end. How do I proceed? I know I am trying to show that the denominator of the sum is even while the numerator is odd.