(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the definition of a limit to prove that lim [(1+a_{n})^{-1}] = 1/2 if lim a_{n}= 1.

2. Relevant equations

([itex]\forall[/itex][itex]\epsilon[/itex]>0)([itex]\exists[/itex]N[itex]\in[/itex]N)(n[itex]\geq[/itex]N [itex]\Rightarrow[/itex]|a_{n}-L|<[itex]\epsilon[/itex])

3. The attempt at a solution

Let [itex]\epsilon[/itex] be arbitrary. Since lim a_{n}exists, [itex]\exists[/itex]N[itex]\in[/itex]Nsuch than |a_{n}-1|<[itex]\epsilon[/itex]'.

My professor helped me a bit, but once we started comparing two different epsilons, I couldn't follow him anymore. He said to choose [itex]\epsilon[/itex]'< 1/2 since 1/2 < a_{n}, but I don't understand why we can say that the sequence is greater than or equal to 1/2 since we only know the value of its limit.

Any help would be appreciated, I've always had a hard time with the rigorous definitions.

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# Homework Help: Definition of a limit of a sequence

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