A U A' means "the element chosen could come from either set 'A' or set 'not A' or both". Since it's impossible to be from both, that means it has to either come from A or not come from it. Thus it could be anything in the universal set.
First show that $A\cup A' \subseteq U$. Then show $U \subseteq A \cup A'$.
Claim: $A \cup A' \subseteq U$
Proof:
Since $A \subseteq U$ so $A' \subseteq U$. Hence, given any $x \in A \cup A'$, at least one of the following is true: $x \in A$ or $x \in A'$. In either case, by the definition of subset, $x \in U$, so $A \cup A' \subseteq U$ as claimed.
Claim: $U \subseteq A \cup A'$
Proof: See post #2 (from Plato) or #5 (from Prove It)