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?
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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.