Cluster point:

**Sheikh_fiji** · March 14th 2009, 12:53 AM

Q: Find a sequence having only one cluster point, yet not convergent.

Solution:
Let N = 1,2,1,3,1,4,1,5,....,1,n,.....

Clearly 1 is a cluster points of N, yet N is unbounded.

Just wondering if this solution is correct? Any suggestions!!!
Thanx

**ThePerfectHacker** · March 14th 2009, 08:59 AM

Originally Posted by Sheikh_fiji

Q: Find a sequence having only one cluster point, yet not convergent.

Solution:
Let N = 1,2,1,3,1,4,1,5,....,1,n,.....

Clearly 1 is a cluster points of N, yet N is unbounded.

Just wondering if this solution is correct? Any suggestions!!!
Thanx

For a sequence $\{a_n\}$ be say that $a$ is an accumulation point if for any $\epsilon > 0$ there exists a distinct $a_N$ so that $|a_N-a| < \epsilon$ . In your example $1$ is not an accumulation point because if $\epsilon = \tfrac{1}{2}$ the only terms of the sequence that are within $\epsilon$ to $1$ is just $1$ itself. It needs to be distinct from $1$ .

**mr. frege** · March 16th 2009, 12:11 PM

...but easily corrected by replacing 1 with any convergent sequence?

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