Unique Cluster Point => Convergent?

**tttcomrader** · September 13th 2008, 05:59 AM

The question is:

Let the sequence $\{ x_n \}$ be in $\mathbb {R}$ 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.

**Plato** · September 13th 2008, 07:06 AM

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

**tttcomrader** · September 13th 2008, 02:03 PM

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 $| t_n - 0 | < \epsilon$ .

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

Is my argument proper? Thank you.

**Plato** · September 13th 2008, 02:21 PM

Originally Posted by tttcomrader

Is my argument proper?

Yes it works.

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