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Let ${a_n}$ be a sequence and $b_n := \frac {2a_n + (-1)^n}{|a_n| +1}$ for $n$ in $\mathbb N$. Prove that ${b_n}$ has a convergent subsequence.

Attempt:

To show that the subsequence converges, I just have to show boundedness right? Is what I have here correct? Am I missing anything?

$|b_n| = |\frac {2a_n + (-1)^n}{|a_n| +1}| = \frac {|2a_n + (-1)^n|}{|a_n| +1} \le \frac {2|a_n| + 1}{|a_n| +1} \le \frac {2|a_n| + 2}{|a_n| +1} = 2$

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  • $\begingroup$ This is OK, if you can use Bolzano-Weierstrass. $\endgroup$ – Ingix Mar 16 '17 at 9:37

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