Let $(\Omega_1, \tau_1)$ and $(\Omega_2, \tau_2)$ be two topological spaces. Prove that $\displaystyle\{{(x_n, y_n) \}}_{n \in \mathbb{N}}$ converges to $(x,y)$ on the product topology if and only if $x_n \to x$ in $\Omega_1$ and $y_ n \to y$ in $\Omega_2$.
My proof:
$\implies$: Let $U \ni x$ and $V \ni y$ be open sets of $\Omega_1$ and $\Omega_2$, respectively. Then $(x, y) \in U \times V$, so by hypothesis there exists $N \in \mathbb{N}$ such that $\displaystyle\{{(x_n, y_n) \}}_{n \in \mathbb{N}} \in U \times V \ \forall n \geq N$. Then $x_n \in U$ and $y_n \in V$ for all but a finite number of terms and we're done.
$\impliedby$: Let $A \ni (x, y)$ be an open set of $\Omega_1 \times \Omega_2$. Clearly there exist $B \in \beta$ and $C \in \mathfrak{C}$ such that $(x,y) \in B \times C \subset A$, where $\beta$ and $\mathfrak{C}$ are basis for $\Omega_1$ and $\Omega_2$, respectively. By hypothesis, for any open sets $\Omega_1 \supset U \ni x$ and $\Omega_2 \supset V \ni y$, there exist $N_1$ and $N_2$ in $\mathbb{N}$ such that $x_n \in U \ \forall n \geq N_1$ and $y_n \in V \ \forall n \geq N_2$. In particular, this is true for $B$ and $C$. Then: $$\displaystyle\{{(x_n, y_n) \}}_{n \in \mathbb{N}} \in B \times C \subset A \ \forall n \geq \max(N_1, N_2).$$
and we're done.
Is all of this alright? It just looks too easy and I wanna make sure I didn't make any mistakes...
The proof I gave is indeed correct.