Prove or disprove the following statements of sequences:
- There is a bounded sequence ${a_n}$ with three limit points -8, 22 and 23.
- There is an unbounded sequence ${a_n}$ with three limit points -8, 22 and 23.
- There is a monotonic sequence ${a_n}$ with three limit points -8, 22 and 23.
- 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.