Q:

Let the limits of the sequences $a_n$ and $b_n$ respectively be $k$ and $k^3.$

I came across the following problem which is as follows:    

Let the limits of the sequences $a_n$ and $b_n$ respectively be $k$ and $k^3.$ If the sequence $a_1,b_1,a_2,b_2,...........$ has a limit,then the value of this limit is which of the following:
  $1.0$ or $1$ or $-1$
  $2.0$ or $1$
  $3.k+k^3$
  $4.k^4.$    

Can someone point me in the right direction?Thanks in advance for your time.

A:

Look if the sequence given by $a_1,b_1,a_2,b_2,...........$ has a limit it implies that all of its converging subsequence must have the same limit.
This implies $a_1,a_2,a_3,...........$ will have the same limit (as this sequence is convergent) as the above sequence and also
$b_1,b_2,b_3,...........$ must also have the same limit (as this sequence is also convergent) as this sequence.
This forces the sequence $a_1,a_2,a_3,...........$ and $b_1,b_2,b_3,...........$ to have the same limit
Implying $k^3=k$
Solving this we get the possible values as $0,1,-1$
This implies that the sequences $a_1,a_2,a_3,...........$ and $b_1,b_2,b_3,...........$ converges to either $0$ or $1$ or $-1$. So the original sequence must also converge to either of the above values(If it converges at all).

