Definition and existence of Riemann integral in PMA Rudin I understand all moments bseides $(1)$ and $(2)$. Why Rudin considers $\inf U(P,f)$ and $\sup L(P,f)$. Why we can't considered $\sup U(P,f)$ and $\inf L(P,f)$?

Can anyone explain it to me please?

• Think about the picture of the upper and lower sums. Clearly you want the lowest possible value (infimum) of the set of upper sums and the supremum of the lower sums. – user137731 Oct 31 '15 at 18:54

This boils down to the fact that if $P'\supseteq P$, then $U(P',f)\le U(P,f)$. So, the finer the partition of $[a,b]$ the smaller the value of $U(P,f)$.
The idea in Riemann integration is to approximate $\int_a^b$ by taking finer and finer partitions of $[a,b]$. So the values $(P,f)$ approach $\int_a^b$ by getting smaller and smaller.
So,if we were to let $\overline\int=\sup$ and $\underline\int=\inf$ the integeral would have little chance of converging.
• Not sure what you're not understanding. It might help if I knew why you think lettting $\overline\int=\sup$ is a good idea. – Tim Raczkowski Oct 31 '15 at 18:50
• I can't understand what problems would be if we let $\overline\int=\sup$ and $\underline\int=\inf$? You wrote that the integral would have little chance of converging. But can it be explained more strictly? – ZFR Oct 31 '15 at 18:58