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Limit of infinite sequence raised to a real number power

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Let {an} be a sequence with positive terms such that lim an = L > 0.

    Prove lim (an)x = Lx.

    2. Relevant equations

    If x is a real number, there exists an increasing rational sequence {rn} with limit x.

    A monotone sequence {an} is convergent if and only if {an} is bounded.

    ax = lim arn.

    3. The attempt at a solution

    I know that each of the individual elements of the sequence {(an)x} converge to (an)x, but I do not know what else I can do from here.

    Any help would be greatly appreciated. Thanks!
     
  2. jcsd
  3. Apr 20, 2010 #2
    It may help to know that every convergent sequence in [itex] \mathBB{R} [/itex] contains a monotone subsequence. The power function [tex] f(x) = x^a [/tex] preserves monotonicity (although it may reverse it for negative powers). What would be a bound for such a sequence? Why would it have to be the least upper bound/greatest lower bound? What does that say about the rest of your sequence?
     
    Last edited: Apr 20, 2010
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