I answerd a question but I feel like there's something missingor wrong.
The question:
Let $a_{n}$ be a sequence with only two partial limits: -1,2.
Let's define a new sequence $b_{n}$:
$b_{n}=\frac{2{a_{n}^{2}-a_{n}-1}}{a_{n}^{2}+1}$.
I need to prove that $b_{n}$ is convergent and to find it's limit.
I think that I should prove that $b_{n}$ is convergent by showing that there is no partial limit other than 1.
so, we can choose $b_{n_{k}}$ that converges to b, so $a_{n_{k}}$ can have only -1 or 2 as a limit, so we can choose $a_{n_{k_{j}}}$ to be convergent subsequence of $a_{n_{k}}$, and to find out no matter what is the limit that $b_{n_{k_{j}}}$ converges to 1, so $b_{n_{k}}$ as well.
Ok, so what's worng? :-)
Thank you.