# liminf of a product

• Dec 30th 2010, 01:57 AM
amheissan
liminf of a product
I am trying to prove the following:
$\underbrace{(\lim \inf f_n)}_f\underbrace{(\lim\inf g_n)}_g\leq \underbrace{\lim\inf (f_ng_n)}_k$.
Here's what I have so far:
FTSOC assume $fg>k$. We know that $f_n and $g_n for only finitely many $n$. Then $f_ng_n for only finitely many $n$. This implies $f_ng_n\geq fg-\epsilon(f+g-\epsilon)$ almost always. So, $f_ng_n\geq k-\epsilon(f+g-\epsilon)$ almost always.
But this doesn't get me anywhere. Where am I going wrong? If I let $\epsilon=\min\{f,g\}$, I just get $f_ng_n\geq 0$. So, this tells me nothing.
Any help would be appreciated. (Even though it doesn't state it, I am assuming that all $f_n, g_n$ are nonnegative.)
• Dec 30th 2010, 08:50 AM
snowtea
I don't think that is true.
Consider $f_n$ the sequence $-1,0,-1,0,...$
and $g_n$ the sequence $2,0,2,0,...$
so $f_ng_n$ is $-2, 0,-2, 0,...$

Now $(\lim \inf f_n)(\lim\inf g_n) = -1 \times 0 = 0$
$\lim\inf (f_ng_n) = -2$
• Dec 30th 2010, 09:59 AM
amheissan
Quote:

Originally Posted by amheissan
...I am assuming that all $f_n, g_n$ are nonnegative.

You are right that it wouldn't hold if were allowed to have a function as you have described, but we are limited to non-negative functions.

Any more thoughts on proving this property? Thanks!
• Dec 30th 2010, 10:18 AM
Drexel28
Quote:

Originally Posted by amheissan
You are right that it wouldn't hold if were allowed to have a function as you have described, but we are limited to non-negative functions.

Any more thoughts on proving this property? Thanks!

Note that by definition $\displaystyle \liminf_{n\to\infty}f_n(x)=\inf \left\{\lim_{n\to\infty}f_{n_k}(x):f_{n_k}(x)\text { is a subsequence of }f_n(x)\right\}$ and similarly for $\displaystyle \liminf_{n\to\infty}g_n(x)$. Use the fact then that $\inf\left(A\right)\inf\left(B\right)\leqslant \inf\left(AB\right)$.
• Dec 30th 2010, 10:33 AM
amheissan
Okay, but how would you prove $\inf(A)\inf(B)\leq \inf (AB)$? I'm really just not seeing this property.
• Dec 30th 2010, 01:17 PM
snowtea
For only nonnegative sequences, Drexel28 is giving you the hint to only look at the inf of the set of values at any subsequence (ignore the order of the sequence).

Let $AB = \{ab | a\in A, b\in B\}$.
Consider any element $a\in A$ and $b\in B$, $ab \geq \inf(A)\inf(B)$ (by definition since everything is nonnegative).

Also, remember that the inf of a subset of AB must be larger than the inf of AB