How to prove that $a
Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that
  $$a<S_n-[S_n]<b$$
  where $[x]$ represents the largest integer not exceeding $x$.
This problem is from China 2012 China Second Round (High school math competition) competition last problem, I think this problem has more nice methods, maybe using analytic methods.

Let $i\geqslant1/(b-a)$ and $k=\lceil S_i\rceil$. Since the sequence $(S_j)_{j\geqslant1}$ is unbounded, some values of this sequence are greater than $k+a$, hence $n=\min\{j\mid S_j\gt k+a\}$ is well defined and finite. Then $S_{n-1}\leqslant k+a\lt S_{n}$ and $n\gt i$ since $S_i\lt k+a$ hence 
$$
S_{n}=S_{n-1}+1/n\lt k+a+1/i\leqslant k+b.
$$
Thus, $k+a\lt S_{n}\lt k+b$, in particular, $\lfloor S_{n}\rfloor=k$ and $a\lt S_{n}-\lfloor S_{n}\rfloor\lt b$.
For every $i$ large enough, this provides some $n\gt i$ such that $a\lt S_{n}-\lfloor S_{n}\rfloor\lt b$. To get infinitely many indexes $n$ such that $a\lt S_{n}-\lfloor S_{n}\rfloor\lt b$, just iterate the construction.
This approach works for every unbounded sequence $(S_n)_n$ such that $S_{n+1}-S_n\to0$.
