Suppose that
\[ (x_\alpha)_{\alpha \in J} \to x \text{ in } X \quad \text{and} \quad (y_\alpha)_{\alpha \in J} \to y \text{ in } Y. \]
Show that \((x_\alpha \times y_\alpha) \to x \times y\) in \(X \times Y\).
Solution
Goal: Prove that if nets \((x_\alpha)_{\alpha \in J}\) and \((y_\alpha)_{\alpha \in J}\) indexed by the same directed set \((J, \succeq)\) converge to \(x \in X\) and \(y \in Y\) respectively, then the product net \((x_\alpha, y_\alpha)_{\alpha \in J}\) converges to \((x, y)\) in the product topology on \(X \times Y\).
1. Setting and open neighborhood reduction:Let \(W \subseteq X \times Y\) be an arbitrary open neighborhood of \((x, y)\).By definition of the product topology, there exist open neighborhoods \(U \subseteq X\) of \(x\) and \(V \subseteq Y\) of \(y\) such that:\[(x, y) \in U \times V \subseteq W.\]
2. Eventual membership in coordinate components:Proof:
2.1. Since \(x_\alpha \to x\) and \(U\) is an open neighborhood of \(x\), there exists \(\alpha_1 \in J\) such that:\[\alpha \succeq \alpha_1 \implies x_\alpha \in U.\]
2.2. Since \(y_\alpha \to y\) and \(V\) is an open neighborhood of \(y\), there exists \(\alpha_2 \in J\) such that:\[\alpha \succeq \alpha_2 \implies y_\alpha \in V.\]
3. Directed set upper bound and product convergence:Proof:
3.1. Since \((J, \succeq)\) is a directed set, there exists an element \(\alpha_0 \in J\) with \(\alpha_0 \succeq \alpha_1\) and \(\alpha_0 \succeq \alpha_2\).
3.2. Let \(\alpha \in J\) satisfy \(\alpha \succeq \alpha_0\).
3.3. By transitivity of the preorder \(\succeq\), \(\alpha \succeq \alpha_1\) (so \(x_\alpha \in U\)) and \(\alpha \succeq \alpha_2\) (so \(y_\alpha \in V\)).
3.4. Thus \((x_\alpha, y_\alpha) \in U \times V \subseteq W\) for all \(\alpha \succeq \alpha_0\).
3.5. Since \(W\) was arbitrary, the net \((x_\alpha, y_\alpha)_{\alpha \in J}\) is eventually in every neighborhood of \((x, y)\), so \((x_\alpha, y_\alpha) \to (x, y)\) in \(X \times Y\). Q.E.D.