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Prove or disprove the following statements of sequences:

  1. There is a bounded sequence ${a_n}$ with three limit points -8, 22 and 23.
  2. There is an unbounded sequence ${a_n}$ with three limit points -8, 22 and 23.
  3. There is a monotonic sequence ${a_n}$ with three limit points -8, 22 and 23.
  4. There is a Cauchy sequence ${a_n}$ with three limit points -8, 22 and 23.

My problem is that I don't really know how to do this...

1) I would say that there is such a bounded sequence,
f.ex. the sequence ${a_n}=(-1)^n$ if $n=2k |\forall k \in \mathbb{N}$then ${a_n}+14$ else
if $n=2k+1$ then ${a_n}*8$ so that there are the lpts (lpt=limit point) -8 and 22 and 23

2)..

3)If ${a_n}$ is monotonic: then $a_n+1>a_n$ or $a_n+1,
so that $a_n = -8$ and $a_(n+1) = 22$ and $a_(n+2) = 23$
so that it works, and there also is such a sequence with these limit points.

4)It is not possible because every real Cauchy sequence converges and therefore only has one limit a, which is also the only limit point.

1 Answers 1

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(1) You’re working much harder than necessary: what’s wrong with $\langle -8,22,23,-8,22,23,\dots\rangle$?

(2) Instead of repeating the block $-8,22,23$ over and over, why not use $-8,22,23,n$, where $n$ is the number of the block?

$\langle-8,22,23,1,-8,22,23,2,-8,22,23,3,-8,22,23,4,-8,22,23,5,\dots\rangle$

(3) No, you can’t make the sequence monotonic and keep all three cluster points. If $-8$ is a cluster point, there are arbitrarily large $n$ with $a_n\in(-9,-7)$, and if $22$ is a cluster point, there are also arbitrarily large $n$ with $a_n\in(21,23)$. Show that $\langle a_n:n\in\Bbb N\rangle$ has a subsequence $\langle a_{n_k}:k\in\Bbb N\rangle$ such that $a_{n_k}\in(-9,-7)$ when $n$ is even, and $a_{n_k}\in(21,23)$ when $n$ is odd, and explain why this subsequence shows that $\langle a_{n_k}:k\in\Bbb N\rangle$ cannot be monotone.

(4) You’re right, and your reasoning is correct.

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    @phil: You’re welcome.2012-12-23