VWO 1990 Problem 4

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function.

(a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\lim_{k\rightarrow +\infty} a_k =+\infty$

(b) Prove that there exist such a sequence ( $a_n$ ) in $\mathbb{R}^+_0$ if you know $\lim_{x\rightarrow +\infty} f(x)=0$ .