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 $c\in C$?

Let $\beta=\sup C$. Then by the definition $$x\le \beta\quad\quad\forall x\in C$$
in particular, $$x\le \beta\quad\quad\forall x\in A$$
So $\beta$ is an upper bound for $A$ and hence $\sup A\le\beta = \sup C$
