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 $\{a_n\}$ 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 $1$.

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