# Limit of infinite sequence raised to a real number power

1. Apr 19, 2010

### issisoccer10

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. Apr 20, 2010

### Hoblitz

It may help to know that every convergent sequence in $\mathBB{R}$ contains a monotone subsequence. The power function $$f(x) = x^a$$ 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