Exercise 26, § 2

(a) Let A and B be two well-ordered subsets of $\mathbf{R}$ (which are necessarily countable; cf. General Topology, IV, § 2, Exercise 1). Show that $A + B$ is well-ordered and that, for all $c \in A + B$, there exists only a finite number of ordered pairs $(a, b)$ such that $a \in A, b \in B$ and $a + b = c$ (to show that a non-empty subset of $A + B$ has a least element, consider its greatest lower bound in $\mathbf{R}$).

(b) Let K be a commutative field. In the vector space $K^\mathbf{R}$, consider the vector subspace E consisting of the elements $(\alpha_x)$ such that the set (depending on $(\alpha_x)$ of $x \in \mathbf{R}$ such that $\alpha_x \neq 0$ is well-ordered. For any two elements $(\alpha_x)$, $(\beta_x)$ of E, we write $(\alpha_x)(\beta_x) = (\gamma_x)$, where $\gamma_x = \sum_{y+z=x} \alpha_y \beta_z$ (a sum which is meaningful by virtue of (a)). Show that this law of composition defines, with addition, a field structure on E; the elements of E are also denoted by $\sum_{t \in \mathbf{R}} \alpha_t X^t$ and called formal power series with well-ordered real exponents, with coefficients in K.*