To prove the Riemann Integrability Criteria:
A bounded function $f$ on the interval [$a, b$] is Riemann integrable iff given an $\epsilon$ > 0 we may determine a positive number $\delta$ so that $\sum_P(sup f-inf f)\Delta x < \epsilon$ for all partitions P whose subintervals have length less than $\delta$.
the book starts with some statements I am not able to follow:
$Proof.$ If we assume that $f$ is Riemann integrable on the interval [$a,b$], then given an $\epsilon> 0$ we have a $\delta > 0$ and a number A so that for any partition of [$a,b$] whose subintervals have length less than $\delta$, every Riemann sum is between $A-\epsilon/4$ and $A+\epsilon/4$. But then the absolute value of the difference of any two Riemann sums is less than $\epsilon/2$, and $\sum_P(sup f -inf f)\Delta x < \epsilon$.
So my questions are:
Where does $\epsilon/4$ come from? Is this choice arbitrary? Can it be $\epsilon/10$ as well?
And then given this initial value, and following with the value of the absolute difference which makes sense, I dont't see then the last statement that goes from the absolute difference to the conclusion that the difference of the Upper and Lower Riemann sums has to be less than $\epsilon$, why not still $\epsilon/2$?