Question 100
Let $S_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a, b$ with $0 \le a \lt b \le 1$, there are infinite many terms in the sequence $\{S_n - [S_n]\}$ that are within $(a, b)$. (Here $[x]$ denotes the largest integer not greater than real number $x$.)