Behavior of the tail of a cdf. If X is an integrable real random variable it is true that
$$ \lim_{x \to \infty} x P(X > x) = 0 \, ?$$ 
I know it is true for the $L^2$ case since it can be derived easily from Chebyshev inequality. Are there any counterexamples?. 

Since $X$ is integrable, 
$$E(X)=\int_{-\infty}^{\infty}\xi\,dP(\xi)<\infty$$ 
exists and is finite and thus also the improper integral
$$S_\infty=\int_{0}^{\infty} \xi\,dP(\xi)<\infty$$
exists and is finite. That means that 
$$S(x)=\int_{0}^{x} \xi\,dP(\xi)$$
converges to $S_\infty$ for $x\to\infty$. Then we get
\begin{align}
\lim_{x\to\infty} x P(X>x)
&=\lim_{x\to\infty} \int_x^\infty x dP(\xi)\\
&\leq\lim_{x\to\infty} \int_x^\infty \xi dP(\xi)\\
&\leq\lim_{x\to\infty} (S_\infty - S(x))=0\\
\end{align}
With the non-negativity of $x P(X>x)$ for $x>0$ the assumption follows.
