> 
> Prove or disprove: $(a\_{n})\_{n\in\mathbb{N}}$ and $(b\_{n})\_{n \in
> \mathbb{N}}\subset \mathbb{R}$ are convergent sequences and the
>  sequence $(c\_{n})\_{n \in \mathbb{N}}$ is defined as
> 
> 
> $$c\_{n}=\left\{\begin{matrix}a\_{n},\text{if n is uneven} & \\ b\_{n},
> \text{if n is even} & \end{matrix}\right.$$
> 
> 
> The sequence $(c\_{n})\_{n\in\mathbb{N}}$ has got two cluster points.
> 
> 
> 

I say the statement is true.

Let $a\_{n}=1^{n}$ and let $b\_{n}=-1^{n}$ ($a\_{n}$ converges to $1$, $b\_{n}$ converges to $-1$)

$\Rightarrow$ $c\_{n}=(-1)^{n}$

which is a well know (divergent) sequence with two cluster points, one is at $1$ and the other one is at $-1$.

I really hope there are no mistakes?