# Spivak - Chapter 8 Problem 3b

Would someone mind clarifying what this question (the one above “4,” preceeded by “b”) is asking and relate the proof of theorem one to the definition of sets A and B? I understand this question may be a bit out of place for the site but it seems like it may be important for the questions which follow and I haven’t been able to unravel its statement.  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?