I am stuck in the following exercice for about a week, I decided to post it to get some help if possible, thanks.
Let $f : [0,+\infty) \to [0,+\infty)$ be a continuous function such that $\lim_{x \to +\infty} f(x)=0$.
Does there exist a convex function $g : [0,+\infty) \to [0,+\infty)$ such that $g \geq f$ and $ \lim_{x \to +\infty} g(x)=0$?
The idea I have, but I am unable to conclude with it, is as follows:
Let $[0,+\infty) =\bigcup_{k} I_{k}$, where $(I_{k})$ a sequence of intervals of length $l_{k} \to 0 $.
We aim to construct a piecewise linear function that satisfies the conditions. Graphically, this seems possible .
Assume that $(n_{i})$ is a strictly increasing sequence such that $n_{1}<n_{1}+1<n_{2}<n_{2}+2<n_{3}<\cdots$ and that $f(x)<\dfrac{1}{i+1}$ for all $x\geq n_{i}$. We let $L_{0}$ the spline joining $(0,M)$ and $(n_{1},1)$ where $M>0$ is large enough such that the spline $L_{1}$ joining $(n_{1},1)$ and $(n_{2},1/2)$ has slope greater than $L_{0}$, we now let $L_{2}$ be the spline joining $(n_{2},1/2)$ and $(n_{3},1/3)$, the splines $L_{n}$ for $n\geq 3$ are defined in the same fashion. Then the function $g=L_{0}\chi_{[0,n_{1}]}+\displaystyle\sum_{i=1}^{\infty}L_{i}\chi_{[n_{i},n_{i+1}]}$ is convex since the slopes of $L_{i}$ are increasing.