Prove that the product of two first countable spaces $X$ and $Y$ is first countable.

Attempt:Let $X$ and $Y$ be two first countable spaces. Let $(x,y) \in X \times Y$. Then $x \in X$ and $y \in Y$. Then $x$ has a countable neighborhood basis, $\beta\_x$.Also $y$ has a countable neighborhood basis $\beta\_y$. Then the set $\beta\_x \times \beta\_y=\{B\_x \times B\_y|B\_x \in \beta\_x,B\_y \in \beta\_y\}$ is a countable neighborhood basis at $(x,y)$. The set is countable since the cardinality is $|\beta\_x \times \beta\_y|$ and the cartesian product of two countable sets is countable. Each basic open set $U \times V$ containing $(x,y)$ contains some $B\_x \times B\_y$, since if $x \in U$ and $y \in V$, there is some $B\_x$ with $x \in B\_x\subset U$ and some $B\_y$ with $y \in B\_y \subset V$.So $(x,y) \in B\_x \times B\_y \subset U \times V$.

Comment: I looked at the proof of this for countable products and noticed the proof to be much more demanding, which makes me unsure if this attempt suffices.However I noticed this is generally the case for proofs involving infinite/finite products. Is this ok, if not how could I change/improve it?Also I am unsure if it appropriate to use the notation $\beta\_x \times \beta\_y$ for the set $\{B\_x \times B\_y|B\_x \in \beta\_x,B\_y \in \beta\_y\}$.