Throughout that question, there is a typo; he's not referring to the proof of theorem $1$, rather the proof of Theorem $7$-$1$. If you refer to the proof of the theorem, from an intuitive level, what Spivak is trying to do with the set $A$ is to locate the smallest $x$ in the interval $[a,b]$ such that $f(x) = 0$. This smallest $x$ turns out to be $\sup A$.
Now, what Spivak wants you to do in $3$(b) is to refer to the same proof (Thm $7$-$1$), and see what happens if you try to mimic that proof, but using the set $B$. If you do so, you will find some point $\xi$ in the interval $[a,b]$ such that $f(\xi) = 0$. Now in general, $x \neq \xi$. So, he wants you to establish a relationship between them; i.e which is larger?