Finding Cluster Points/Accumulation Points

  • MHB
  • Thread starter brooklysuse
  • Start date
  • Tags
    Points
In summary, the set of cluster points for the set A := {(−1)n/n : n ∈ N} is 0. This can be proven using the definition of a cluster point and showing that 0 is the only point that satisfies the criteria.
  • #1
brooklysuse
4
0
Find the set of cluster points for the set A := {(−1)n/n : n ∈ N}. Justify your answer with
proof.

I believe 0 is a cluster point but I can't figure out how to prove this, or how to prove any other point is not.

Any quick help would be appreciated. Thanks.
 
Physics news on Phys.org
  • #2
brooklysuse said:
Find the set of cluster points for the set A := {(−1)n/n : n ∈ N}.
I believe 0 is a cluster point
Hmm, I believe \(\displaystyle \frac{(-1)n}{n}=-1\).
 
  • #3
brooklysuse said:
Find the set of cluster points for the set A := {(−1)n/n : n ∈ N}. Justify your answer with
proof.

I believe 0 is a cluster point but I can't figure out how to prove this, or how to prove any other point is not.

Any quick help would be appreciated. Thanks.
I guess you mean $A := \{(−1)^n/n : n \in \Bbb{N}\}$. You are correct that $0$ is the only cluster point. To prove it, you will need to use the definition of a cluster point, which is ... ? (Start from there.)
 

FAQ: Finding Cluster Points/Accumulation Points

What is a cluster point/accumulation point?

A cluster point, also known as an accumulation point, is a point in a set of numbers where an infinite number of points in the set are arbitrarily close to it. In other words, any neighborhood around the cluster point will contain an infinite number of points from the set.

How do you find cluster points/accumulation points?

To find cluster points, you must first determine the set of numbers you are working with. Then, you can use the definition of a cluster point to identify points in the set that are arbitrarily close to each other. These points will be the cluster points of the set.

What is the difference between a cluster point and a limit point?

While both cluster points and limit points are points in a set that are arbitrarily close to each other, there is a subtle difference between the two. A cluster point must have an infinite number of points in the set that are arbitrarily close to it, while a limit point only needs to have at least one point in the set that is arbitrarily close to it.

Can a set have more than one cluster point/accumulation point?

Yes, a set can have multiple cluster points or accumulation points. This is because there can be multiple points in a set that are arbitrarily close to each other, making them all cluster points. However, there can only be one limit point for a set.

How are cluster points/accumulation points used in mathematics?

Cluster points and accumulation points are important concepts in real analysis and topology. They are used to study the behavior of sequences and series, as well as to define the notions of continuity and compactness. They are also useful in proving theorems and solving problems in these areas of mathematics.

Similar threads

I Question about Accumulation points
Replies
11
Views
3K
MHB The Closure of a Set is Closed .... Lemma 1.2.10
Replies
2
Views
2K
MHB Finding border points and then interior points
Replies
2
Views
2K
Verification that ZxZ has no cluster points in RxR
Replies
3
Views
1K
I Discrete Subrings of Complex Numbers: Topology of Rings
Replies
11
Views
2K
I If a sequence converges, then all subsequences of it have same limit
Replies
7
Views
3K
I Proving a Fixed Point Theorem for Shrinking Maps on Compact Spaces
Replies
8
Views
3K
I On limit of convolution of function with a summability kernel
Replies
2
Views
1K
MHB Prove that series converges uniformly
Replies
3
Views
1K
MHB Infimum and Supremum of a Set (Need Help Finding Them)
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top