Exercise 1, § 2

Let $E$ be a linearly ordered set with a smallest element $\omega$, and let $I$ be an interval of $E$, containing $\omega$, with $\omega$ as its left-hand end-point. Suppose that $E$ and $I$ satisfy axioms (GR_I), (GR_{II}), (GR_{IV}) and the following axiom:

(GR_{IIIb}) The set of elements $x > \omega$ in $I$ is not empty, and if $x$ is any element $> \omega$ in $I$, there exists $y > \omega$ in $I$ such that $y^2 \leq x$.

Show that there exists an increasing mapping $f$ of $I$ into $\mathbf{R}$, such that $f(\omega) = 0$ and $f(xy) = f(x) + f(y)$ whenever $x, y$ and $xy$ are in $I$; also that $f(I) \cap [0, f(b)]$ is dense in $[0, f(b)]$ for all $b \in I$.