# Thread: liminf of a product

1. ## liminf of a product

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.)

2. I don't think that is true.
Consider $\displaystyle f_n$ the sequence $\displaystyle -1,0,-1,0,...$
and $\displaystyle g_n$ the sequence $\displaystyle 2,0,2,0,...$
so $\displaystyle f_ng_n$ is $\displaystyle -2, 0,-2, 0,...$

Now $\displaystyle (\lim \inf f_n)(\lim\inf g_n) = -1 \times 0 = 0$
$\displaystyle \lim\inf (f_ng_n) = -2$

3. Originally Posted by amheissan
...I am assuming that all $\displaystyle f_n, g_n$ 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!

4. 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 $\displaystyle \displaystyle \liminf_{n\to\infty}f_n(x)=\inf \left\{\lim_{n\to\infty}f_{n_k}(x):f_{n_k}(x)\text { is a subsequence of }f_n(x)\right\}$ and similarly for $\displaystyle \displaystyle \liminf_{n\to\infty}g_n(x)$. Use the fact then that $\displaystyle \inf\left(A\right)\inf\left(B\right)\leqslant \inf\left(AB\right)$.

5. Okay, but how would you prove $\displaystyle \inf(A)\inf(B)\leq \inf (AB)$? I'm really just not seeing this property.

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

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