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Give an example of a sequence $\{a_n\}_{n\ge 1}$ which has no limit, but such that the set of real numbers $E=\{a_n:n\ge 1\}$ has a unique limit point.

Any help to this problem will be appreciated.

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HINT: Try a sequence whose odd-numbered terms are all the same and whose even-numbered terms converge to some other number.

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A different hint: Begin with any convergent sequence, and ,,spoil'' it in sufficiently many places to ensure it does not converge, but take care not to create additional limit points that way. You could for instance take the convergent sequence $c_n := \frac{1}{n}$ and define: $$ a_n = \begin{cases} n & \text{if $n$ is a power of 2} \\ \frac{1}{n} & \text{if $n$ is not a power of $2$} \end{cases} $$

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