a_n has no convergent subsequences => lima_n = infinity

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November 3rd 2009, 12:47 PM
Jazz10

a_n has no convergent subsequences => lima_n = infinity

Prove that if a_n is a sequence that contains no convergent subsequences, the lim abs(a_n) = infinity
November 3rd 2009, 01:05 PM
Plato

Quote:

Originally Posted by Jazz10

Prove that if a_n is a sequence that contains no convergent subsequences, the lim abs(a_n) = infinity

Can you prove that each sequence contains a monotone subsequence?
But any bounded monotone sequence converges.
So if you prove the first then you have solved this problem.

Here is a start. Define $S = \left\{ {k:\left( {\forall n > k} \right)\left[ {a_n > a_k } \right]} \right\}$
Two cases: $S$ is infinite or $S$ is finite.
Case one gives a increasing subsequence.
Case two gives a decreasing subsequence.
November 3rd 2009, 02:21 PM
Jazz10

I appreciate the help and see where you are going with it, however, can you clarify what you mean by each seqeuence? as in each subsequence has a monotone subsequence itself? not really sure how that ties in to your hint at the end
November 3rd 2009, 03:04 PM
Plato

Quote:

Originally Posted by Jazz10

I appreciate the help and see where you are going with it, however, can you clarify what you mean by each seqeuence? as in each subsequence has a monotone subsequence itself? not really sure how that ties in to your hint at the end

Frankly, after I posted that I realized a simpler proof.
Define $\left( {\forall N \in \mathbb{Z}^ + } \right)\left[ {S_N = \left\{ {a_k :\left| {a_k } \right| \leqslant N} \right\}} \right]$ .

What if some $S_J$ were infinite?
Would we have a convergent subsequence? WHY?

One real problems with help-sites such as this is simply that helpers have no idea what theorems the poster knows and therefore can use.
November 3rd 2009, 03:38 PM
Jazz10

ooo, nice, so using your setup, if we had an set Sj that were infinite, then that infinite set would be bounded and would therefor have an accumulation point, and thus we could find subsequence that was convergent. Since there are no convergent subsequences, there cannot exist a j that bounds any set, and therefore the sequence must diverge to inifity or negative infinity.

is that where you were going?