Is liminf of a product equal to the product of liminfs?

Question

My question is just for curiosity. I was thinking if is true this curious affirmation:

Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then $\lim \inf (a_n b_n) = (\lim \inf a_n) (\lim \inf b_n) $ This is true?

For $b_n\ge 0$ see: http://math.stackexchange.com/questions/275124/checking-of-a-solution-to-how-to-show-that-lim-sup-a-nb-n-ab and http://math.stackexchange.com/questions/510314/proof-that-limsup-a-nb-n-limsup-a-n-lim-b-n — Martin Sleziak, Aug 16 '14 at 05:08

score 2 · Accepted Answer · answered Aug 16 '14 at 01:57

2

Let $b_n=-1$ for all $n$. Say $a_n$ alternates between 1 and 3. Then $(\lim\inf b_n)(\lim\inf a_n)=-1$, but $\lim\inf a_n b_n=-3$.

answered Aug 16 '14 at 01:57

mathematician

2,450

What do you think if $a_n$ is convergent? – math student Aug 16 '14 at 02:26
@mathstudent If both sequences are convergent, their termwise product is also convergent. – Chris Culter Aug 16 '14 at 02:46
You don't even need to go to negative numbers to get this result... – Thomas Andrews Aug 16 '14 at 02:57
oh you're right guys, sorry. I was sleepy when I writed the question. Thanks for your attentions! – math student Aug 16 '14 at 03:36

Is liminf of a product equal to the product of liminfs?

1 Answers1