Question 11
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 $f(n+1)\gt f(n)$ infinitely often.
b. Prove that $f(n+1)\lt f(n)$ infinitely often.
$$
f(n)=\frac{1}{n}\left(\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor\right),
$$
where $\lfloor x\rfloor$ denotes the integer part of $x$.
a. Prove that $f(n+1)\gt f(n)$ infinitely often.
b. Prove that $f(n+1)\lt f(n)$ infinitely often.