Does Lim Inf of Product Equal Product of Lim Infs for Non-Negative Sequences?

[jostpuur]

If f_1,f_2,f_3,\ldots and g_1,g_2,g_3,\ldots are some arbitrary real sequences, is it true that

<br /> \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) = (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n)?<br />

For arbitrary \epsilon &gt;0 there exists N\in\mathbb{N} so that

<br /> n &gt; N\quad\implies\quad f_n &gt; \underset{k\to\infty}{\textrm{lim inf}}\; f_k \;-\; \epsilon,<br />

so

<br /> \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) \geq \underset{n\to\infty}{\textrm{lim inf}}\big((\underset{k\to\infty}{\textrm{lim inf}}\; f_k \;-\; \epsilon) g_n\big) \;=\; (\underset{k\to\infty}{\textrm{lim inf}}\; f_k \;-\; \epsilon) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n),<br />

which implies

<br /> \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) \geq (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n),<br />

but I don't know how to do the other direction.

edit: I just realized I'm assuming \underset{n\to\infty}{\textrm{lim inf}}\; g_n \geq 0 in the calculation, although it was not my original intention, but I think I'll try to not fix it in the remaining editing time. (Actually assuming \underset{k\to\infty}{\textrm{lim inf}}\; f_k - \epsilon \geq 0 too...)

edit edit: In fact I think I'll add the assumption that the sequences are non-negative, because otherwise I have a counter example (f_n)_{n\in\mathbb{N}} = (1,-1,1,-1,\ldots), (g_n)_{n\in\mathbb{N}}=(-1,1,-1,1,\ldots).

 

[jostpuur]

I think I got this thing solved. If f_1,f_2,f_3,\ldots and g_1,g_2,g_3,\ldots are some arbitrary non-negative sequences, then

<br /> \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) \geq (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n),<br />

is the best one can say about the liminfs. One counter example for other direction is

<br /> (f_n)_{n\in\mathbb{N}} = (1,\frac{1}{2},1,\frac{1}{2},\ldots),<br /> \quad\quad<br /> (g_n)_{n\in\mathbb{N}} = (\frac{1}{2},1,\frac{1}{2},1,\ldots),<br />

because

<br /> \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) = \frac{1}{2} &gt; \frac{1}{4} = (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n).<br />

Thank you very much.