In Spivak's calculus book he provides a proof for: Theorem: If f is continuous on [a, b] and f(a) < 0 < f(b), then there is some number x in [a, b] such that f(x) = 0. In the proof he explicitly says, "...A has a least upper bound [itex]\alpha[/itex] and that a < [itex]\alpha[/itex] < b. We now wish to show that f([itex]\alpha[/itex]) = 0, by eliminating the possibilities f([itex]\alpha[/itex]) < 0 and f([itex]\alpha[/itex]) > 0." This doesn't seem to apply as intuitively to me. How can a least upper bound be in the middle of a set? If we describe x to be between values a and b, then shouldn't the least upper bound and greatest lower bound be b and a, respectively? Also, he continues to explain the first case, where f([itex]\alpha[/itex]) < 0. He says, "There is a [itex]\delta[/itex] > 0 such that f(x) < 0 for [itex]\alpha[/itex] - [itex]\delta[/itex] < x0 < [itex]\alpha[/itex]." I'm confused with the notation here. I'm unsure what [itex]\delta[/itex] represents, so if someone can offer an alternative explanation of this segment that would be greatly appreciated. Thank you! Also, the theorem is intuitive to me. I understand that if f is continuous and a < 0 and b > 0 then there has to be a point where it crosses the x-axis, but of course the proof is of more value here than the intuitive sense of the problem.