Thread: Can a sequence has more than one accumulation point?

1. Can a sequence has more than one accumulation point?

Let a seqence be in a metric space, can it has more than one accumulation point?

My answer is no, because if it has more than one, then it would converge to both points, implies they are the same. Would that be the right start?

2. How are you handling divergent sequences?

3. I actually need to know the same answer. I was thinking that a sequence could have more than one accumulation point. For example the sequence {(-1)^n}? I may be wrong though. If there is more than one accumulation point, how would I go about proving that?

4. Originally Posted by EricaMae
I actually need to know the same answer. I was thinking that a sequence could have more than one accumulation point. For example the sequence {(-1)^n}? I may be wrong though. If there is more than one accumulation point, how would I go about proving that?
You are correct.

The sequence $\displaystyle x_n = \left( { - 1} \right)^n + \frac{1}{n}$ has two accumulation points, $\displaystyle \{1,-1\}$

5. I found these pages in my textbook, I think it's what you're talking about....

It starts on the first page and then it's the top of the second page.

6. Originally Posted by Showcase_22
I found these pages in my textbook, I think it's what you're talking about....

It starts on the first page and then it's the top of the second page.
An accumulation point of a squence is not the same thing as the limit of the sequence. An accumlation point may be taken to be defined as a limit of a sub-sequence, or alternativly as point such that every open interval containing the point also contains an element of the sequence.

See Plato's post for an example.

7. oops sorry.

Could this adapted somehow to prove that a sequence can have more than one accumulation point?

I am aware that the proof in the textbook proves a sequence cannot have more that one limit, so I am doubtful, but it would be great if it could provide a starting point.

8. Originally Posted by Showcase_22
oops sorry.

Could this adapted somehow to prove that a sequence can have more than one accumulation point?

I am aware that the proof in the textbook proves a sequence cannot have more that one limit, so I am doubtful, but it would be great if it could provide a starting point.
Plato's example is sufficient to show this.

RonL

9. If you deal with a bounded sequence which does not have limit.
Then this means that this sequence have more than one convergent subsequences.
Refer the example given by Plato (see also Bolzano?Weierstrass theorem - Wikipedia, the free encyclopedia).