Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a convex and integrable function such that $\lim_{x\to \infty} f(x) - x$ exists and is finite. I am also given the fact that $\int_{0}^1f(x)dx=\frac{1}{2}$. I have to show that for any $n$ a natural number, $\int_{0}^1x^nf(x)dx \le \frac{1}{n+2}$.
I thought that the easiest way would be to show that $f(x)\le x$ for every $x$ in between $0$ and $1$, but I have no clue how should I use the fact that $f(x) - x$ has a limit at $+\infty$.