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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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On the proof of Theorem 15.2 in Montgomery-Vaughan's Multiplicative number theory

I am trying to understand the proof of Theorem 15.2 in the aforementioned book. In this proof, the authors seem to infer that, if $\psi(x) - x < x^{\Theta - \epsilon}$ for every $\epsilon > 0$ and su …