1. ## divergent sequences

Question:
A sequence diverges if it has a subsequence which diverges.

Is this statement true? Is there a counterexample?
(Intuitively I would say this is false.)

Thanks.

2. ## Re: divergent sequences

Originally Posted by CountingPenguins
Question:
A sequence diverges if it has a subsequence which diverges.
Can you prove this: If a sequence converges the any of its subsequences converge to the same limit.

3. ## Re: divergent sequences

Yes, is the converse true?

4. ## Re: divergent sequences

Originally Posted by CountingPenguins
Yes, is the converse true?
Not the converse, the contrapositive is true.
If a subsequence diverges then the sequence is divervent.

5. ## Re: divergent sequences

True or False, if true prove, if false show a counterexample.

A sequence $s_n$ diverges if it has a subsequence which diverges.

6. ## Re: divergent sequences

Originally Posted by CountingPenguins
True or False, if true prove, if false show a counterexample.
A sequence $s_n$ diverges if it has a subsequence which diverges.
If a sequence converges then any subsequence of that sequence also converges.

The contrapositive which is also true is:
If a sequence diverges then then any super-sequence of that sequence also diverges.

7. ## Re: divergent sequences

I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.

I just keep thinking there is a counterexample to this.

Is it a fact that if there is a divergent subsequence the whole sequence diverges?

8. ## Re: divergent sequences

Originally Posted by CountingPenguins
I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
I just keep thinking there is a counterexample to this.
Is it a fact that if there is a divergent subsequence the whole sequence diverges?
Do you understand what the exact definition of sub-sequence means?

If you do, how is the a counter-example to anything?

9. ## Re: divergent sequences

Originally Posted by CountingPenguins
I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
I just keep thinking there is a counterexample to this.
Is it a fact that if there is a divergent subsequence the whole sequence diverges?
Do you understand what the exact definition of sub-sequence means?

If you do, how is the a counter-example to anything?

10. ## Re: divergent sequences

Originally Posted by CountingPenguins
I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
I just keep thinking there is a counterexample to this.
Is it a fact that if there is a divergent subsequence the whole sequence diverges?
Do you understand what the exact definition of sub-sequence means?

If you do, how is this a counter-example to anything?

11. ## Re: divergent sequences

I finally see what you have been saying. Thanks.