I am trying to prove the following:

$\displaystyle \underbrace{(\lim \inf f_n)}_f\underbrace{(\lim\inf g_n)}_g\leq \underbrace{\lim\inf (f_ng_n)}_k$.

Here's what I have so far:

FTSOC assume $\displaystyle fg>k$. We know that $\displaystyle f_n<f-\epsilon$ and $\displaystyle g_n<g-\epsilon$ for only finitely many $\displaystyle n$. Then $\displaystyle f_ng_n<fg-\epsilon(f+g-\epsilon)$ for only finitely many $\displaystyle n$. This implies $\displaystyle f_ng_n\geq fg-\epsilon(f+g-\epsilon)$ almost always. So, $\displaystyle f_ng_n\geq k-\epsilon(f+g-\epsilon)$ almost always.

But this doesn't get me anywhere. Where am I going wrong? If I let $\displaystyle \epsilon=\min\{f,g\}$, I just get $\displaystyle f_ng_n\geq 0$. So, this tells me nothing.

Any help would be appreciated. (Even though it doesn't state it, I am assuming that all $\displaystyle f_n, g_n$ are nonnegative.)