My question is about a "small" technicality regarding taking the infimum/minimum in the following proof for Riemann integrability.
The claim to prove is the following: Suppose $f: [a, b] \to \mathbb{R}$ is bounded. Then $f$ is Riemann integrable if for every $\epsilon > 0$ there exists a partition $P$ of $[a, b]$ s.t. $U(P, f) - L(P, f) < \epsilon$, where $U(P, f)$ and $L(P, f)$ denote the upper and lower Darboux sums of $f$ over the partition $P$.
Outline of the proof: Let $IU(P, f)$ and $IL(P, f)$ denote the upper and lower Darboux integrals of $f$ over the partition $P$, respectively. Since the claim is logically: $\forall \epsilon > 0. \exists \text{ a partition P over } [a, b]. U(P, f) - L(P, f) < \epsilon \implies IU(P, f) = IL(P, f)$, one can use contrapositive to prove instead that $\exists \epsilon > 0. \forall \text{ a partition P over} [a, b]. IU(P, f) > IL(P, f) \implies U(P, f) - L(P, f) \geq \epsilon$.
Now here begins my question: Since we are only interested in such partitions where $IU(P, f) > IL(P, f)$, we can and should choose the smallest possible $\epsilon$ such that $U(P, f) - L(P, f) \geq \epsilon$. Specifically we should choose $\epsilon = U(P, f) - L(P, f)$ where $\epsilon$ is the smallest available, i.e. $P$ minimizes the equation $U(P, f) - L(P, f)$. By definition the smallest possible value of $U(P, f) - L(P, f)$ is given by $\epsilon = \inf \{U(P, f) - L(P, f)\mid \text{P is a partition over } [a, b]\}$. But now, could it be possible to construct a real number sequence from the partitions of $[a, b]$ so that the infimum of the aforementioned set is equal to zero? Isn't this a problem as $\epsilon > 0$? Moreover isn't it true that we cannot use the minimum, as there is no guarantee that the minimum is contained in the said set? What is the correct way to argue that if such a partition $P$ can be found for every positive epsilon, then necessarily the upper and lower Darboux integrals are equal, i.e. $f$ is Riemann integrable?
I wanted to leave this as another comment, but I think it's too long. When you say "Since we are only interested in such partitions where (,)>(,)", this doesn't make sense!
The statement you are trying to prove is $$\forall \epsilon\exists P_{[a,b]}\big( U(P_{[a,b]},f) - L(P_{[a,b]},f) < \epsilon \big) \implies f \text{ is integrable } \iff IU(P,f) = IL(P,f) $$
(note this means your statement of the contrapositive is incorrect due to the quantifiers; they do not apply to the RHS)
This is simply notation, but note that on the LHS statement, I write $P_{[a,b]}$, and for the Darboux integrals I write $P$. Note that $IU(P,f) = \text{inf} \{U(P,f): P \text { is a partition of }[a,b]\}$, and $P$ here is a bound variable. But $P_{[a,b]}$ refers to a specific partition. To prove the theorem then you must use this specific partition that you are given exists, and show that the values of the upper and lower integrals are equal.
A hint to complete the proof: $IU(P,f) \leq U(P,f)$, for any partition $P$ (think about the definition of the upper integral). A similar inequality exists for the lower integral. Try using these to complete the proof!