Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

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

share|improve this answer

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: $$ a_n = \begin{cases} n & \text{if $n$ is a power of 2} \\ \frac{1}{n} & \text{if $n$ is not a power of $2$} \end{cases} $$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.