Exercise 6, § 5

Let $X_1$ (resp. $X_2$) be a set filtered by a filter $\mathfrak{F}_1$ (resp. $\mathfrak{F}_2$) and let $f$ be a real-valued function defined on $X_1 \times X_2$. Show by an example that the three numbers $\lim \sup_{\mathfrak{F}_1 \times \mathfrak{F}_2} f(x_1, x_2)$, $\lim \sup_{\mathfrak{F}_1} (\lim \sup_{\mathfrak{F}_2} f(x_1, x_2))$, $\lim \sup_{\mathfrak{F}_2} (\lim \sup_{\mathfrak{F}_1} f(x_1, x_2))$ are in general distinct.