Integrability proof

Question

I have trouble understanding the fact that

as a result of the fact that

These statements come from the proof

Answer 1

Since $n$ is arbitrary, we can choose it to be as large as we want, such that we may choose $\frac{M}{n} mE$ to be as small as we want. Since $\inf \int_E \psi(x)dx - \sup \int_E \phi(x)dx$ must now be smaller than something that we may choose to be arbitrarily small, it must be equal to $0$ by the Archimedean property of the real numbers.