Synopsis
Please verify my proof. I would also appreciate any tips on how I might improve my mathematical writing. Thank you.
If my proof is without major issues, please note that I might delete the question soon afterwards.
Exercise
If every member of a set $\mathscr{A}$ has cardinality $\kappa$ or less, then $$\text{card } \bigcup \mathscr{A} \leq (\text{card } \mathscr{A}) \cdot \kappa$$
Proof
Let $K$ be a set of cardinality $\kappa$. Consider the function $F: \bigcup \mathscr{A} \rightarrow A \times K$ defined by $F(x) = (A_x, g_x(x))$ where $A_x$ is the corresponding set in $\mathscr{A}$ such that $x \in A_x \subseteq \bigcup \mathscr{A}$ and $g_x$ is an injection from $A_x$ into $K$ guaranteed by our initial conditions. Then $F$ describes a suitable injection from $\bigcup \mathscr{A}$ into $\mathscr{A} \times K$ and $\text{card} \bigcup \mathscr{A} \leq (\text{card } \mathscr{A}) \cdot \kappa$.
Note: I was wondering if the "choosing" of an injection $g_x$ requires me to state that I am utilizing the Axiom of Choice
Your function $F$ is not well-defined: what if $x \in A$ and $x \in B$ for $A\neq B \in \mathscr{A}$? You might be able to solve this by assuming the sets in $\mathscr{A}$ are disjoint and then use something like $$\left|\bigcup_{i \in I} A_i\right| \le \left|\bigcup_{i \in I} A_i \times \{i\}\right|$$
Via cardinal arithmetic, an easy proof is the following:
$$\left|\bigcup \mathscr{A} \right| \le \sum_{A \in \mathscr{A}}|A| \leq \sum_{A \in \mathscr{A}} \kappa = |\mathscr{A}| \kappa $$