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!

If $x\in A \cup B $ then either:


*

*$x\in A$ 


or


*

*$x\in B$ 


If $x\in A$ then by the hypothesis $x\in C$. The same for the case for which $x\in B$. 
So, in any case $A\cup B\subseteq C$.
