Deduce that if $A$ is a subset of $C$, then $\sup A\leq \sup C$. 
Deduce that if $A$ is a subset of $C$, then $\sup A\leq \sup C$.

How do I begin with the above proof?
I can't see why $\sup A$ must either be equal of less than $\sup C$.
Is it not possible for some $a \in A$ to be greater than some $c\in C$?

Hint: Start with an upper bound of one of the two sets, and try to show that it is also an upper bound of the other. (You'll have to figure out which is which.)
