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Suppose the image of some sequence has 1 limit point.

Convergent sequence, for example $a_n = \frac1n$, may have 1 limit point. I wonder, is there any divergent sequence with 1 limit point?

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How about $0,1,0,2,0,3,0,4,...$ –  copper.hat Jul 2 '13 at 6:21
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2 Answers 2

Exactly what do you mean by limit point? If you mean a point that is the limit of the sequence, then of course a divergent sequence of real numbers has no limit point. If you mean a cluster point, however, then the answer is yes: an example is the sequence

$$\langle x_n:n\in\Bbb N\rangle=\langle 0,1,0,3,0,5,0,7,0,9,\dots\rangle$$

in which

$$x_n=\begin{cases} 0,&\text{if }n\text{ is even}\\ n,&\text{if }n\text{ is odd}\;. \end{cases}$$

This is a divergent sequence with a single cluster point, $0$.

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Damn, you are fast. –  copper.hat Jul 2 '13 at 6:22
    
@copper.hat: Had a typo, though. (I remember thinking the same thing about Arturo. Frequently.) –  Brian M. Scott Jul 2 '13 at 6:27
    
Still amazing. ${}{}$ –  copper.hat Jul 2 '13 at 6:29
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How about

$$ a_n = \begin{cases} 0, & n \text{ even}\\ n, & n \text{ odd} \end{cases}$$

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