I was wondering how to finish this proof, or if I'm even on the right track.
Let $b=\sup Q$. Then $x\leq b$ $ \forall x \in Q$ and $b \leq$ any upper bound of $Q.$
Since $P \subset Q, \\ \forall y \in P, y \in Q$
This means that $y \leq b$
I know that $\sup P \in Q$ since all of the elements in $P$ are in $Q$. Since any element of $P$ must be less than or equal to b, can I say that $\sup P$ must be less than or equal to $\sup Q$?