Question 27
For a natural number $n$ let $$f(n)=\frac{1}{n} \sum_{k=1}^{n}\left\lfloor\frac{n}{k}\right\rfloor.$$ Prove that there are infinitely many natural numbers $m$ for which the inequality $f(m)\lt f(m+1)$ holds, and that there are infinitely many natural numbers $m$ for which the inequality $f(m)\gt f(m+1)$ holds.