I am currently on chapter 8 of Spivak's Calculus and I'm struggling to understand the reason behind some of the proof of "Theorem 7-1" (IVM). Why has the variable $x_0$ been introduced (what is the utility of it)?
Could you not just say that $x_1$ is a number such that $a < x_1 < a + \delta$ and state the contradiction?
The problem with your suggestion is that you need to prove that $x_1\in A.$ All you know a priori is that $f$ is negative on $(\alpha - \delta, x_1).$ The role of $x_0$ in the proof is to bridge the gap between $a$ and $x_1,$ by the equation $[a, x_1]=[a,x_0]\cup[x_0,x_1].$