# Limit of infinite sequence raised to a real number power

## 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

Related Calculus and Beyond Homework Help News on Phys.org
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?

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