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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.

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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?

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