So I was attempting to derive the exponentiation rules for real exponents, taking as a starting point the exponentiation rules for rational exponents. I began by defining $a^x = \lim_{n \rightarrow +\infty} a^{x_n}$, where $a, x$ are real numbers (with $a > 0$), and $x_n$ is a sequence of rational numbers that converges to $x$.
Now, to show that this operation is well-defined, I considered another sequence, $x_n'$, that also converges to $x$, and tried to show that $\lim_{n \rightarrow +\infty} a^{x_n} = \lim_{n \rightarrow +\infty} a^{x_n'}$, by showing that:
\begin{equation} \lim_{n \rightarrow +\infty} \frac{a^{x_n}}{a^{x_n'}} = 1 \end{equation}
From the properties of rational exponentiation, I can rewrite the quotient as $a^{x_n - x_n'}$, and of course that intuitively, one can see that as $n$ gets ever larger, $x_n$ and $x_n'$ get ever closer, and so $a^{x_n - x_n'}$ can get arbitrarily close to $1$.
My question, though, is: how would one prove this formally? I.e., let us write the limit condition ($\varepsilon$ is real, $p, n$ are a positive integers):
\begin{equation} \forall \varepsilon > 0 \; \exists p \; \colon n \ge p \Rightarrow \lvert a^{x_n - x_n'} - 1 \rvert < \varepsilon \end{equation}
How would one, given an $\varepsilon$, proceed to produce a $p$ such that for $n\ge p$, the consequent of the implication holds?
Thank you in advance for your time.