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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