where is my mistake in this liminf innquality proof

**transgalactic** · January 9th 2009, 04:14 AM

i need to proove that lim inf(An*Bn)>=liminfAn * liminfBn

i got the opposite resolt
where is my mistake:
http://img187.imageshack.us/img187/6705/44112683ho1.gif

**Opalg** · January 9th 2009, 05:12 AM

Here's an example where the inequality is strict. You can test your argument against this example, to pinpoint where you went wrong.

Let $a_n = 2+(-1)^n$ , $b_n = 2-(-1)^n$ . Thus a_n is alternately 1 and 3, b_n is alternately 3 and 1. The lim inf for both sequences is 1. But when you multiply the sequences together you get a sequence where each term is 3. So the lim inf of the product is 3.

So $\liminf_{n\to\infty}a_nb_n = 3 > 1 = \liminf_{n\to\infty}a_n\liminf_{n\to\infty}b_n$ .

**transgalactic** · January 9th 2009, 05:50 AM

i cant see the mistake
i am done this proof strictly on this definition

http://img515.imageshack.us/img515/5666/47016823jz1.gif

**Opalg** · January 9th 2009, 10:29 AM

Originally Posted by transgalactic

i cant see the mistake
i am done this proof strictly on this definition

http://img515.imageshack.us/img515/5666/47016823jz1.gif

As far as I can tell from your work, you are claiming that if $b = \liminf_{n\to\infty} b_n$ , then $b\geqslant b_n$ for all n. That is completely untrue.

**transgalactic** · January 9th 2009, 11:13 AM

why its not true??

in each of the following sequence our temporary inf
gets bigger or stays the same

so the function of the inf's goes up

and their limit is the least upper bound(b)

and the least upper bound is bigger or equal than every member
in the sequence.

can you tell me whats wrong with what i said?

**Opalg** · January 9th 2009, 11:43 AM

Originally Posted by transgalactic

why its not true??

in each of the following sequence our temporary inf
gets bigger or stays the same Correct.

so the function of the inf's goes up Correct.

and their limit is the least upper bound(b) It is the least upper bound of that sequence of temporary infs. It is not the least upper bound of the original sequence.

and the least upper bound is bigger or equal than every member
in the sequence. So the lim inf is greater than or equal to each of the numbers $\color{red}\inf\{a_k:k\geqslant n\}$ . That does not make it greater than or equal to a_n.

can you tell me whats wrong with what i said?

..

**transgalactic** · January 9th 2009, 12:22 PM

ok the limit is bigger or equal then any temporary inf of this sequence.
so what could i say the relations of the lim inf with the members of the sequence.
here i was given a solution

http://www.mathhelpforum.com/math-he...-question.html

and i was told that liminf is smaller than any temporary inf.

how he came to this conclusion??

**transgalactic** · January 9th 2009, 12:24 PM

what is the relations of liminf with the members of a sequence

??

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