(1) In this question, you'll prove that the exponential is a valid operation for real numbers and prove some of its properties. For all parts, fix $x > 1$.
(a) For $y \in \mathbb{R}$, define the set
$E(x, y) = \{x^t \mid t \le y, t \in \mathbb{Q}\}$.
(Recall that $x^{\frac{p}{q}} = (x^{\frac{1}{q}})^p$ for any $p \in \mathbb{Z}$, $q \in \mathbb{N}$.) Suppose that $y$ is rational. Prove that $x^y = \sup E(x, y)$.
(b) For $y \in \mathbb{R}$ (not necessarily rational), show that $E(x, y)$ is bounded.
(c) Define $x^y = \sup E(x, y)$. Prove that for $x > 1$, $y, z \in \mathbb{R}$, $x^{y+z} = x^y x^z$. Explain why this implies that the function $f: \mathbb{R} \to \mathbb{R}_{>0}$ given by $f(y) = x^y$ is injective.
