I am currently going through the book Topics in Algebra and Analysis, and am unable to understand the solution to the following problem on page 7,
Exercise 1.5. Prove that there are an infinite no. of pairs of irrational no.s $a,b$ such that $a + b = ab$ is an integer number.
I understood the first half of the solution where they found the expressions for $a$ and $b$, but I was unable to understand their proof for the irrationality of the discriminant. Why exactly do we have to take $n \geq 5$, and how do we get the inequality immediately after that? I feel like they skipped a few 'obvious' steps which aren't obvious to me. Besides, is there a simpler way to prove this? I have searched elsewhere but found nothing, so I would be grateful if anyone could help me understand this proof.