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## 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

^{rn}.

## 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!