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Thread: Relations between similar limits?

  1. #1
    Junior Member
    Nov 2008

    Relations between similar limits?

    Sorry about the vague title. I've made apost similar to this... Here it goes:

    Give an epsilon - N proof that:

    if a[n] -> a then a[n]^2 -> a^2

    and the same for

    if a[n] -> a then a[n]^4 -> a^4

    Any Ideas? Again, cheers
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  2. #2
    Super Member Gamma's Avatar
    Dec 2008
    Iowa City, IA
    Let $\displaystyle \epsilon > 0$ be given.
    By assumption $\displaystyle a_n \rightarrow a < \infty$.

    Choose N such that for all n>N
    $\displaystyle |a_n-a|<\frac{\epsilon}{2a}$
    Now for the real N, we may need to go bigger depending on the epsilon, we must consider |M| is the maximum value of the sequence past N, at some point the sequence must be bounded if it converges to a. The point is we need to choose N so that $\displaystyle |a_n-a|<min\{\frac{\epsilon}{2a}, \frac{\epsilon}{2|M|}\}$ for all n>N.

    $\displaystyle |a_n^2 - a^2|= |a_n^2 - a_na + a_na -a^2| \leq ||a_n||a_n-a| + |a||a-a_n|< |M|\frac{\epsilon}{2|M|} + a\frac{\epsilon}{2a}=\epsilon$

    I think you can figure out the second part, remember
    $\displaystyle a^4=(a^2)^2$
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