Harmonic Fractional Parts Are Dense

Let $S_n = 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n}$ for positive integers $n$. Prove that for any real numbers $a, b$ with $0 \leqslant a < b \leqslant 1$, infinitely many terms of the sequence $\left\{S_n - \lfloor S_n \rfloor\right\}$ belong to $(a, b)$. (Here $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$.)