A problem with limsup/liminf

Hi !

This is a proof my teacher wanted us to do, but when coming to giving the solution, he got confused (Worried)

The problem is :

$\displaystyle \left(\liminf_{n \to \infty} f_n(x)\right)^-=\limsup_{n \to \infty} \left(f_n(x)\right)^-$

Where $\displaystyle .^-$ is 'the negative part of the function' :

$\displaystyle f(x)=(f(x))^+-(f(x))^-$

$\displaystyle |f(x)|=(f(x))^++(f(x))^-$

etc...

Also, $\displaystyle f_n$ is a sequence of measurable functions, but I'm not sure it intervenes here.

As a sidenote, this was used in a proof of the generalization of Beppo-Levi's lemma.

Thanks in advance !