let $b$ be a real number, and let $b_1, b_2, b_3, \dots$ be an infinite sequence of rational numbers such that the sequence $(b_n)$ has limit $b$. It can be shown that the sequence $(a^{b_n})$ has a limit, which is independent of the particular sequence $(b_n)$ that we have chosen, as long as the sequence has $b$ as a limit.
Then for $a\gt 0$ we can define $a^b$ as the limit of the sequence $(a^{b_n})$.
For $r ,s$ irrational and $a\gt0$ I want to prove $$ (a^{r}) ^s=a^{rs}$$
Let $a_n\to r$ and $b_m \to s$ are sequence of rational. then $$ (a^{r}) ^s=\lim_{m \to \infty} \left \{ \left(\lim_{n \to \infty} a^{a_n} \right)^{bm} \right\}$$
Now from just first step I don't know what to do .Any hints on this ?