Exercise 5, § 4

As the point $(x, y)$ tends to $(+\infty, -\infty)$ while remaining in $\mathbf{R} \times \mathbf{R}$, shows that the set of cluster points of the function $x + y$ is $\overline{\mathbf{R}}$.

(*) Use the property of uniqueness, up to isomorphism, of the maximal ordered algebraic extension of an ordered field [B. L. van der Waerden, Moderne Algebra, vol. 1, 1st edition (Berlin, Springer, 1930) pp. 232-234].

Likewise, as $(x, y)$ tends to $(0, +\infty)$ while remaining in $\mathbf{R} \times \mathbf{R}$, the set of cluster points of $xy$ is $\overline{\mathbf{R}}$.