Proof of Riemann's Criterion for Integrability, and the Integral Lies between Lower and Upper Sums

Let $\mathcal{L}(f)$ and $\mathcal{U}(f)$ be the set of lower sums and the set of upper sums of $f$, and let $\lambda=\sup\mathcal{L}(f)$ and $\upsilon=\inf\mathcal{U}(f)$ be the lower and the upper integral of $f$ over $[a,b]$, The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iii); they exist and are the numbers $\lambda_{*}$ and $\lambda^{*}$ of Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (iv), so $\lambda\le\upsilon$. By The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii), the elements of $\mathcal{L}(f)$ and $\mathcal{U}(f)$ are the sums $L(f,P)$ and $U(f,P)$, $P$ ranging over the partitions of $[a,b]$. As $\lambda$ is an upper bound of $\mathcal{L}(f)$ and $\upsilon$ a lower bound of $\mathcal{U}(f)$, by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv),

$$L(f,P)\le\lambda\le\upsilon\le U(f,P)\qquad\text{for every partition }P\text{ of }[a,b].\tag{1}$$

Proof of (ii). If $f$ is integrable, then $\int_{a}^{b}f=\lambda=\upsilon$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), and (1) gives $L(f,P)\le\int_{a}^{b}f\le U(f,P)$.

Proof of (i). Suppose that for every real $\varepsilon>0$ there is a partition $P$ with $U(f,P)-L(f,P)<\varepsilon$. Let $\varepsilon>0$ and choose such a $P$. By (1), $0\le\upsilon-\lambda\le U(f,P)-L(f,P)<\varepsilon$. As $\varepsilon>0$ was arbitrary, $\upsilon-\lambda=0$ by Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (iii), so $\lambda=\upsilon$ and $f$ is integrable on $[a,b]$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv).

Conversely, let $f$ be integrable, so $\lambda=\upsilon=\int_{a}^{b}f$, and let $\varepsilon>0$. The set $P_{0}=\{a\}\cup\{b\}$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (iii) and Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v), and $a,b\in[a,b]$ (Basic Facts about Intervals of the Real Line (v)), so it is a partition of $[a,b]$ by Partitions of a Closed Interval, the Set of Partitions, and Refinements (i); thus $L(f,P_{0})\in\mathcal{L}(f)$ and $U(f,P_{0})\in\mathcal{U}(f)$, and by Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (iii) $U(f,P_{0})$ is an upper bound of $\mathcal{L}(f)$ and $L(f,P_{0})$ a lower bound of $\mathcal{U}(f)$. The real number $\varepsilon/2$ satisfies $0<\varepsilon/2$ and $\varepsilon/2+\varepsilon/2=\varepsilon$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vii)). First, as $\mathcal{L}(f)$ is nonempty and bounded above, Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (v), applied with $\varepsilon/2$, gives $s\in\mathcal{L}(f)$ with $\lambda-\varepsilon/2<s$, and by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i) $s=L(f,P_{1})$ for a partition $P_{1}$ of $[a,b]$; so $\lambda-\varepsilon/2<L(f,P_{1})$. Then, as $\mathcal{U}(f)$ is nonempty and bounded below, Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (vii), applied with $\varepsilon/2$, gives $s'\in\mathcal{U}(f)$ with $s'<\upsilon+\varepsilon/2$, and by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii) $s'=U(f,P_{2})$ for a partition $P_{2}$ of $[a,b]$; so $U(f,P_{2})<\upsilon+\varepsilon/2$. Let $P=P_{1}\cup P_{2}$, a partition of $[a,b]$ refining $P_{1}$ and $P_{2}$ by The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (v). By Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (ii), $L(f,P_{1})\le L(f,P)$ and $U(f,P)\le U(f,P_{2})$. Hence

$$U(f,P)-L(f,P)\le U(f,P_{2})-L(f,P_{1})<\Big(\int_{a}^{b}f+\frac{\varepsilon}{2}\Big)-\Big(\int_{a}^{b}f-\frac{\varepsilon}{2}\Big)=\varepsilon .$$