# Limit point and cluster point

• Apr 1st 2013, 01:34 AM
rokman54
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.
• Apr 1st 2013, 02:47 AM
Plato
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.
• Apr 1st 2013, 04:07 AM
rokman54
Re: Limit point and cluster point
What is the formal definition of cluster point?
• Apr 1st 2013, 04:10 AM
Plato
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.
• Apr 1st 2013, 05:22 AM
HallsofIvy
Re: Limit point and 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.