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
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 $\displaystyle 1$.