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"Prove that $ \lim_{n \to \infty} \, \, \left(\frac{1}{1+a_n} \right) = \frac{1}{2}$ if $\lim_{n \to \infty} a_n = 1$."

I understand the algebra, but when I get to this step:

$ |1-a_n|\left| \frac{1}{2(1+a_n)} \right| < \epsilon $

I have no idea what to do. Am I allowed to just divide the right-sided product to the epsilon side? Likewise, I don't think I can bound the right-sided product, since if $-1 < a_n < -0.5$ it explodes. I am so close to getting what I want, but don't know how to get there. Any help please?

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    Remember you need to make use of the fact that $a_n \rightarrow 1$. Since $a_n \rightarrow 1$, given $\epsilon > 0$, there exists $N$ such that $\forall n > N$, $\left|a_n - 1 \right| < \epsilon$. Make use of this and also bound $\left| \frac1{2(1+a_n)} \right|$ by $1$ (Why?).2012-03-14

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