Rudin proceeds to prove this as follows:
Given $\epsilon>0, \exists$ a partition P such that $$U(P,\alpha')-L(P,\alpha') < \epsilon$$ Let $M=sup|f(x)|$
He then proves that $$ |U(P,f,\alpha)-U(P,f\alpha')| \leq M\epsilon$$
Also, he has shown that the above is true for any refinement of $P$ as well.
It is clear to me till here. But then he 'concludes' that
I fail to understand how this follows from the the fact that the inequality for $P$ is also true for any refinement of $P$.