Let $a>0$, and let $x$ be a real number. Prove that if $\{r_{n} \}$ is any decreasing rational sequence with limit $x$, then $a^x = \lim_{n \rightarrow \infty} a^{r_n}$
Where in the book $a^x$ is defined as $\lim_{n \rightarrow \infty} a^{s_n}$ where $ \{ s_n \}$ is an increasing sequence.