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- Apr 13, 2013

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Let the sequence $(a_{n})$ with $a_{1}>0$ and $a_{n+1}=1+\frac{2}{1+a_{n}}$.Show that the subsequences $a_{2k}$ and $a_{2k-1}$ are monotonic and bounded.Find the limit $\lim_{n \to \infty} a_{n}$,if it exists.

Do I have to show separately that the two subsequences are monotonic and bounded??Or is there an other way to show it??Could I for example show that $a_{n}$ is monotonic and bounded??