ISL 2006 N3

The sequence $f(1), f(2), f(3), \ldots$ is defined by $$f(n)=\frac{1}{n}\left(\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor\right),$$ where $\lfloor x\rfloor$ denotes the integer part of $x$. (a) Prove that $f(n+1)>f(n)$ infinitely often. (b) Prove that $f(n+1)<f(n)$ infinitely often. (South Africa)