# Thread: Unique Cluster Point => Convergent?

1. ## Unique Cluster Point => Convergent?

The question is:

Let the sequence $\displaystyle \{ x_n \}$ be in $\displaystyle \mathbb {R}$ that has a unique cluster point, does that implies the sequence converges?

I was looking up the theorems, and I believe it is false since we did not say that the sequence is bounded. But are there any examples for that?

Thanks.

2. Consider the sequence $\displaystyle t_n = \left\{ {\begin{array}{ll} n & {\mbox{n odd}} \\ {\frac{1}{n}} & {\mbox{n even}} \\ \end{array} } \right.$
Does it have a unique cluster point?

3. Yes, 0 is the unique cluster point, since for any natural N, I can find all even number of n greater than or equal N such that $\displaystyle | t_n - 0 | < \epsilon$.

And this sequence doesn't converge, since if I let $\displaystyle \epsilon = 1/2$, let $\displaystyle N \in \mathbb {N}$, then whenever $\displaystyle n \geq N$, I would have $\displaystyle |t_n| > 1/2$ when n is odd.

Is my argument proper? Thank you.