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Limit of infinite sequence raised to a real number power

  • #1

Homework Statement



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

Prove lim (an)x = Lx.

Homework 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.

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!
 

Answers and Replies

  • #2
11
0
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?
 
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