# Thread: Cluster point:

1. ## Cluster point:

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

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

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