Misunderstanding in Spivak's Proof of the Intermediate Value Theorem

Solution 1:

Short answer: Spivak is writing a textbook, so he is being rather verbose, and dotting the i's and crossing the t's.

Long answer: you asked

Isn't it true that, because $\alpha$ is a least upper bound of $A$ and $f$ is continuous on $[a,b]$, it must be true that if $a\leq x <\alpha$ then $f(x)<0$ ?

Yes, it is true. But how do you know it is true? You either have to demonstrate that it is true, or appeal to a previously proven Theorem. (Just "intuitively this must be the case" is not "rigorous" enough in mathematics!) In this case, Spivak showed, as part of that last paragraph, precisely that claim. The choice of $x_0$ is to justify your quote above.

And yes, you can also obtain the same conclusion by arguing via contradiction, as you indicated at the end of your post.