1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook