Sequence: product of sequences diverges

[tarheelborn]

1. Homework Statement

Sorry, I posted this earlier but I had an error in my problem statement; please advise. Thank you.

If a_n diverges to +inf, b_n converge to M>0; prove a_n*b_n diverges to +inf

2. Homework Equations



3. The Attempt at a Solution

My attempt follows: I seem to have trouble getting things in the right order, so I am trying to work on my technique, with your help. Also, I am afraid I may have omitted reference to some theorem that I am taking for granted, which is another of my bad habits. Please review for me and advise as appropriate. I am determined to conquer this subject! Thanks.

Let M, e > 0, M, e \in R. By definition of a limit of a sequence, we can choose N_b such that |b_n - M|< e, n >= N. Then -e < b_n - M < e, so M - e < b_n < M + e. So b_n > M - e. We can then choose N_a such that a_n >= M/(M-e), n >= N. Let N = max (N_b, N_a). Then a_n*b_n >= M/(M-e), n >= N. Thus, {a_n*b_n} diverges to + infinity.

 

[Mark44]

When you show that a sequence diverges to infinity, you don't use epsilon; you show that starting with some index in the sequence, all elements are larger than some (large) number M.

The definition for such a sequence goes something like this:
For all numbers M > 0, there exists an index N such that, if n >= N, a_n > M.

You already know that a_n gets large without bound. What can you say about a_n*b_n? Make sure that you don't use M for both a_n and a_n*b_n.

 

[tarheelborn]

It seems as if I should be able to say a_n*b_n gets (large without bound * L), where L = lim b_n, but I don't know how to say that. Instinctively I know that this happens.

 

[tarheelborn]

What if I say:

--------------------------------------------------------------------------------

1. Homework Statement

If a_n diverges to +inf, b_n converge to 0; prove a_n*b_n diverges to +inf

2. Homework Equations



3. The Attempt at a Solution

My attempt follows: I seem to have trouble getting things in the right order, so I am trying to work on my technique, with your help. Also, I am afraid I may have omitted reference to some theorem that I am taking for granted, which is another of my bad habits. Please review for me and advise as appropriate. I am determined to conquer this subject! Thanks.

Let L, M, e > 0, L, M, e \in R. By definition of a limit of a sequence, we can choose N_b such that |b_n - L|< e, n >= N. Then -e < b_n - L < e, so L - e < b_n < L + e. So b_n > L - e. We can then choose N_a such that a_n >= M/(L-e), n >= N. Let N = max (N_b, N_a). Then a_n*b_n >= [M/(L-e)]*L-e, which = M, n >= N. Thus, {a_n*b_n} diverges to + infinity.

 

[Mark44]

What if I say:

--------------------------------------------------------------------------------

1. Homework Statement

If a_n diverges to +inf, b_n converge to 0; prove a_n*b_n diverges to +inf

You've changed the problem on me! In the first post in this thread, b_n converged to a positive number.

Which is it?

 

[tarheelborn]

It does converge to a positive number. SORRY. I posted this twice, once had an error and I seem to be picking up the wrong one somehow. The problem is: if a_n diverges to +infinity and b_n converges to M>0, prove a_n*b_n converges to +infinity.

 

[Mark44]

OK, that's better. Now go back and read what I wrote in post #2. Epsilon doesn't enter into things at all when your trying to show that a sequence diverges to infinity. You can't get within epsilon of infinity, and that seems to be what you're trying to do.

 

[tarheelborn]

I understand that I need to say that a_n*b_n > M, therefore a_n*b_n diverges. But don't I have to initially deal with epsilon in the sense that the initial sequence b_n DID converge to a limit? I was trying to get that limit canceled out so that the "result" was that the product is greater than my M. I am not trying to be stubborn; I just don't get it. The only thing I get in class is the proof copied from my book onto the chalkboard. It just hasn't sunk in yet.

 

[Mark44]

Let L, M, e > 0, L, M, e \in R. By definition of a limit of a sequence, we can choose N_b such that |b_n - L|< e, n >= N. Then -e < b_n - L < e, so L - e < b_n < L + e. So b_n > L - e. We can then choose N_a such that a_n >= M/(L-e), n >= N. Let N = max (N_b, N_a). Then a_n*b_n >= [M/(L-e)]*L-e, which = M, n >= N. Thus, {a_n*b_n} diverges to + infinity.

On a closer look, and with your later explanation, this looks pretty good. I don't see anything wrong in your argument.

I got confused when you changed the limit of your converging sequence from a positive number to 0, which completely changes the outcome.

One thing I would suggest is to spread things out a bit to improve readability. What you have is very dense, making it more difficult to comprehend.

 

[tarheelborn]

Thank you so much! I will definitely try to work on making things more readable. I really appreciate your help.