Exercise 1, § 5

Let $X_1, X_2$ be two directed sets and let $f$ be a real-valued function defined on $X_1 \times X_2$ such that, for each $x_1 \in X_1$, the mapping $x_2 \to f(x_1, x_2)$ is increasing on $X_2$, and for each $x_2 \in X_2$, the mapping $x_1 \to f(x_1, x_2)$ is increasing on $X_1$. Show that $f$ has a limit in $\overline{\mathbf{R}}$ with respect to the product of the section filters of $X_1$ and $X_2$, and that this limit is the least upper bound of $f$.