Limit point and cluster point

My professor vaguely defined a cluster point and I'm a bit confused on the difference between limit point and cluster point.

He said that

" Suppose a_n converges to alpha.

Sequence {a_n} has a cluster point, if a subsequence of a_n converges to alpha.

If a is a cluster point, it is a limit point."

Can you clarify the difference between cluster point and limit point.

I am currently in a introductory analysis class.

Thank you.

Re: Limit point and cluster point

Quote:

Originally Posted by

**rokman54** My professor vaguely defined a cluster point and I'm a bit confused on the difference between limit point and cluster point. He said that

" Suppose a_n converges to alpha.

Sequence {a_n} has a cluster point, if a subsequence of a_n converges to alpha.

If a is a cluster point, it is a limit point."

Can you clarify the difference between cluster point and limit point.

I am currently in a introductory analysis class.

Look at this webpage.

There is a very common misconception here. The limit of a sequence is a cluster point of the sequence but a cluster of a sequence may not be a limit of a sequence. The sequence $\displaystyle \left( {{{\left( { - 1} \right)}^n} + \frac{1}{n}} \right)$ has two cluster points, $\displaystyle 1~\&~-1$ but the sequence does not converge so it has no limit although it does have two limit points.

Re: Limit point and cluster point

What is the formal definition of cluster point?

Re: Limit point and cluster point

Quote:

Originally Posted by

**rokman54** What is the formal definition of cluster point?

Did you look at that web page? *Cluster point* is just another type of *limit point*.

Re: Limit point and cluster point

The Wikipedia page Plato links to gives the definition of a "cluster point":

A point x ∈ X is a cluster point or accumulation point of a sequence $\displaystyle (x_n)$, n ∈ N, if, for every neighbourhood V of x, there are infinitely many natural numbers n such that $\displaystyle x_n$ ∈ V.

It also says "The set of all cluster points of a sequence is sometimes called a limit set." implying that "cluster point" and "limit point" are names for the same thing. However, Plato's point, before, was that "cluster point" or "limit **point**" of a sequence may not be a **limit** of that sequence.