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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.

  • 0
    What is the definition of partial or sub sequential limit?2011-04-27

1 Answers 1

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I propose that you first prove (from your favourite definition of "partial limit") that the sequence $a_n$ can be split into two convergent sequences without overlap or rest.

The proof for $b_n$ will then be much simpler to write down.