Limit of infinite sequence raised to a real number power

[issisoccer10]

Homework Statement

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

Prove lim (a_n)^x = L^x.

Homework Equations

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

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

a^x = lim a^r_n.

The Attempt at a Solution

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

Any help would be greatly appreciated. Thanks!

 

[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?