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# limits watch

1. if the limit of sequence a is infinity, and the limit of sequence b is c, how do i show the limit of sequence ab is infinity? also c>0
2. (Original post by asdfyolo)
if the limit of sequence a is infinity, and the limit of sequence b is c, how do i show the limit of sequence ab is infinity? also c>0
There are a few different ways to do it I should imagine, some brief thoughts on the subject - you'll want to be more rigorous than I have been.
A brief sketch is that converges so it is eventually close to .
Since this is the case, and you know that is divergent, you can see that, for large the sequence and so must diverge. If you can express that more carefully, then you are done.

Bigger hint in the spoiler - try to prove it yourself first. [spoiler]Eventually you have .
3. An alternative approach: You may already have done some of these steps, in which case you can skip them.

1) Prove that if , a sequence diverging to , then .
2) Prove that if is a sequence diverging to infinity, and is a sequence such that for all sufficiently large , then .
3) Since , there is in particular some in the naturals such that for all , . Use this, and the above, to prove the result.
4. (Original post by joostan)
There are a few different ways to do it I should imagine, some brief thoughts on the subject - you'll want to be more rigorous than I have been.
A brief sketch is that converges so it is eventually close to .
Since this is the case, and you know that is divergent, you can see that, for large the sequence and so must diverge. If you can express that more carefully, then you are done.

Bigger hint in the spoiler - try to prove it yourself first. [spoiler]Eventually you have .
Spoiler:
Show
This holds for sufficiently large.
Spoiler:
Show
You'll now want to use the fact that - remember what this means.
It is NOT true that eventually b_n - epsilon > c. Did you mean b_n > c - epsilon?
5. (Original post by DFranklin)
It is NOT true that eventually b_n - epsilon > c. Did you mean b_n > c - epsilon?
Yup, been having problems with spoilers, had to type it out twice - errors must've crept in.
Thanks

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