Prove $A \cup B \subseteq C$ using element chasing Question:

Give an element chasing proof that for any sets $A$, $B$, and $C$: if $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$. Proof should use case analysis.

I need some help proving this  using element chasing and case analysis. I'm not entirely sure what it means by using case analysis, but I understand how to do basic element chasing proofs.
I am unsure as to how to start this proof because of the required guidelines needed. I might be overthinking it as I usually do, but any push in the right direction is always helpful!

Given some $x \in A \cup B$, there are two cases. It might be the case that $x \in A$ and since $A \subseteq C$ this implies $x \in C$. Otherwise we have $x \in B$ and since $B \subseteq C$ this also implies $x \in C$. Therefore $A \cup B \subseteq C$.
