So far I have this
Suppose A, B, and C are any set where A $\subseteq$ B.
Proof that A $\cup$ C $\subseteq$ B U C: (Show that x $\in$ B$\cup$C)
Let x $\in$ A $\cup$ C and then by definition of union x $\in$ A or x $\in$ C. So by definition of subset then x $\in$ B. Thus by definition of union x $\in$ B $\cup$ C as was to be shown.
So basically I was wondering do I need two cases for when x is an element of a or b or is this fine? Also I guess with set proof what justifies more than one case?