Exercise 2, § 1

An algebraic number is a complex number which is algebraic over the field $\mathbf{Q}$ of rational numbers. Show that the ordered field $B$ of all real algebraic numbers is a maximal ordered field, but that it is not complete in the topology induced by that of $\mathbf{R}$ [this topology is the same as $\mathscr{T}_0(B)$; cf. Chapter IV, § 3, Exercise 2 b)].