Expectation: Is this proposition true or false?

[LoadedAnvils]

If X is a continuous random variable and E(X) exists, does the limit as x→∞ of x[1 - F(x)] = 0?

I encountered this, but so far I have neither been able to prove this, nor find a counterexample. I have tried the mathematical definition of the limit, l'Hopital's rule, integration by parts, a double integral (through expectation), and various proof scribbles, but so far, nothing has worked. Can anyone help me with this?

EDIT: In this case, the function F is the CDF of X.

 

[economicsnerd]

A hint, for the case where $X\geq 0:$

$$\mathbb E X = \int_0^\infty xf(x)dx \geq \int_0^{\bar x} xf(x)dx + \bar x\int_{\bar x}^\infty f(x)dx.$$

Think about the pieces of that, and think about limits as $\bar x \to \infty.$

 

[LoadedAnvils]

I'm aware of that, and I know how to prove it for the case of $X \geq 0$, but I'm confused about the case of the entire real line.

Also, what do you mean by $\overline{x}$?

 

[economicsnerd]

I was just using $\bar x$ as another stand-in variable.

If you know how to prove it for nonnegative-valued $X$, then you're basically done. The limiting property you care about is the same for $X_+ = \text{max}\{X,0\}$.

 

[LoadedAnvils]

What do you mean by limiting property?

 

[LoadedAnvils]

Oh! I get it now! Thank you so much!