I am studying Riemann integral from Denlinger's Elements of Real Analysis.
Consider a set $A$ and $B$ subset of $\mathbb R$. $\;\forall a\in A,\forall b\in B, a\leq b$.
As $A$ is bounded above by elements of $B$, thus by completeness property, $\exists sup(A)\in\mathbb R$.
Similarly, $\exists inf(B)\in\mathbb R$.
$$\text{If}\;sup(A)=inf(B)=u,\;then\;\forall\epsilon>0,\exists a\in A\;and\;b\in B\;s.t.\;(b-a)<\epsilon$$
It is easy to show.
By $\epsilon$-criterion of supremum and infimum,
$\exists a\in A,\;s.t.\;u-\epsilon/2<a<u$
Similarly $\exists b\in B, s.t.\;u<b<u+\epsilon/2$
Thus, $b-a<\epsilon$.
Now coming to the integration.
Consider a function $f$ defined and bounded over [a,b]. $P$ is an n-partition of $[a,b]$.
$\underline{S}(f,P)=\sum_im_i\Delta_i,\;where\;m_i=\text{inf}f[x_{i-1},x_i]$
$\overline{S}(f,P)=\sum_iM_i\Delta_i,\;where\;M_i=\text{sup}f[x_{i-1},x_i]$
$A={\{\underline{S}(f,P):\;\text{P is a partition of [a,b]}}\}$
$B={\{\overline{S}(f,P):\;\text{P is a partition of [a,b]}}\}$
We can see that $\forall a\in A,\forall b\in B, a\leq b$.
Thus $sup(A)$ and $inf(B)$ exists.
$sup(A)=\underline{\int_a}^b f$ is lower Darboux integral.
$inf(B)=\overline{\int_a^b} f$ is upper Darboux integral.
Thus, $\underline{\int_a}^b f\leq\overline{\int_a^b} f$
A function $f$ is said to be integrable over $[a,b]$ in $\underline{\int_a}^b f=\overline{\int_a^b} f=\int_a^b f$
Riemann criterion for integrability
A bounded function $f:[a,b]\to\mathbb R$ is integrable on $[a,b]$ iff
$$\boxed{\forall\epsilon>0,\exists\;\text{a partition}\;P\;of\;[a,b]\;s.t.\;\overline{S}(f,P)-\underline{S}(f,P)<\epsilon}$$
We can see that the above definition easily follows from the concepts of supremum and infimum of sets as we have seen in the first paragraph.
I have a doubt that, why in the definition of Riemann criterion only single partition is considered.
We have seen that the $sup(A)=inf(B)$.
So, $\exists a\in A,\;b\in B,\;s.t.\;(b-a)<\epsilon$.
It might happen that this $a$ and $b$ does not correspond to a single partition $P$ but two different partitions $P_!$ and $P_2$.
Why only single partition is included in the definition?
I think that the definition of Riemann criterion should be
$$\forall\epsilon>0,\exists\;\text{atleast two partitions}\;P_1\;and\;P_2\;of\;[a,b]\;s.t.\;\overline{S}(f,P_2)-\underline{S}(f,P_1)<\epsilon$$
I want to know that is my reasoning correct? Isn't the definition of Riemann criterion for integrability given in the book less precise?
Please tell whether am I correct or not?
If there are partitions $P_1$ and $P_2$ such that $\overline S(f,P_2)-\underline S(f,P_1)<\varepsilon$ and if $P=P_1\cup P_2$, then $\overline S(f,P)-\underline S(f,P)<\varepsilon$, since $\overline S(f,P)\leqslant\overline S(f,P_1)$ and $\underline S(f,P)\geqslant\underline S(f,P_1)$.