How to prove: Any positive real number power of positive real number is positive.

Question

I have a question regarding how to prove: Any positive real number power of a positive real number is positive.

I have shown that any positive rational power of positive real numbers is positive. I know the proof has something to do with the denseness of Q in R. Can someone help me?

Answer 1

Assuming $c^x$ where $c\in\mathbb{R}^+, x\in\mathbb{R}$ is defined as $$c^x = \lim_{r(\in\mathbb{Q})\to x} c^r$$ the proof that any positive rational power of a positive real number is positive can be trivially extended to real powers. (If $x+\delta$ and $x-\delta$ are both positive rationals for an arbitrarily small $\delta$, $x$ must be positive.)