Why does x > a in this proof?

Question

The proof for Theorem 13.9 from "Introduction to Metric & Topological Spaces" by Wilson Sutherland. The definition for compact is that any open cover can be reduced to a finite open cover for a compact set.

https://books.google.dk/books?id=dxrpDAAAQBAJ&pg=PA129&lpg=PA129&dq=Wilson+A.+Sutherland+Compactness+of+closed+bounded+intervals&source=bl&ots=-M-q4YnqKS&sig=ACfU3U0nKuavUPNxcnTqBZYWVA7kdJjX1g&hl=de&sa=X&ved=2ahUKEwji0t7ZrIbnAhXqtYsKHXBuDRIQ6AEwAHoECAkQAQ#v=onepage&q&f=false

I understand the proof, but I don't see why we need x > a.