Proof of A Continuous Function on a Closed Interval is Riemann Integrable

Proof of (i). By The Real Line: Standing Notation and Background for Calculus on Intervals (x) (Extreme Value Theorem on a Closed Real Interval (i)) there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$, so $f$ is bounded and its upper sums and lower sums are defined. By The Real Line: Standing Notation and Background for Calculus on Intervals (x), $[a,b]$ is compact (Closed Interval $[a,b]$ is Compact in $\mathbb{R}$ (i)), so $f$ is uniformly continuous on $[a,b]$ by Heine-Cantor Theorem: Continuity on a Compact Metric Space or a Compact Subset Implies Uniform Continuity (ii); in the form of The Real Line: Standing Notation and Background for Calculus on Intervals (vi): for every real $\eta>0$ there is a real $\delta>0$ such that all $s,t\in[a,b]$ with $|s-t|<\delta$ satisfy $|f(s)-f(t)|<\eta$.

Let $\varepsilon>0$ be real. We choose, in this order, $\eta$, $\delta$, $N$ and a partition $P$, and show $U(f,P)-L(f,P)<\varepsilon$. Put $\eta=\varepsilon/(2(b-a))$, which is positive as $b-a>0$. Choose $\delta>0$ for this $\eta$ as above. By The Real Line: Standing Notation and Background for Calculus on Intervals (vii) (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix)) choose $N\in\mathbb{N}$ with $(b-a)/\delta<N$. As $0<N$ in $\mathbb{R}$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi)), $0<N^{-1}$ by Rules of Arithmetic and Order in an Ordered Field (viii). Multiplying $(b-a)/\delta<N$ by $\delta>0$ gives $b-a<N\delta$, and multiplying this by $N^{-1}$ gives $h=(b-a)/N<\delta$, both by Rules of Arithmetic and Order in an Ordered Field (vi).

The uniform partition. Let $P=\{a+(k-1)h:k\in[N+1]\}$. By The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ix), $P$ is a partition of $[a,b]$, the $n$ of The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (i) for $P$ is $N$, its increasing listing is $x:[N+1]\to P$, $k\mapsto x_{k}=a+(k-1)h$, and $x_{i+1}-x_{i}=h$ for every $i\in[N]$.

The oscillation on each subinterval. Let $i\in[N]$ and $s,t\in[x_{i},x_{i+1}]$. Then $s-t\le x_{i+1}-x_{i}$ and $-(x_{i+1}-x_{i})\le s-t$, by Rules of Arithmetic and Order in an Ordered Field (iv) and Rules of Arithmetic and Order in an Ordered Field (v), so $|s-t|\le x_{i+1}-x_{i}=h$ by Rules of Arithmetic and Order in an Ordered Field (x), and $|s-t|<\delta$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (i). As $x_{i},x_{i+1}\in P\subseteq[a,b]$ and $[a,b]$ is an interval (Basic Facts about Intervals of the Real Line (v)), $[x_{i},x_{i+1}]\subseteq[a,b]$ by Basic Facts about Intervals of the Real Line (ii); so $s,t\in[a,b]$, and the choice of $\delta$ gives $|f(s)-f(t)|<\eta$, in particular $|f(s)-f(t)|\le\eta$.

The criterion. As $\eta\ge0$, Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound (iv) gives

$$U(f,P)-L(f,P)\le\eta\,(b-a)=\varepsilon/2<\varepsilon,$$

the last step by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vii). As $\varepsilon>0$ was arbitrary, $f$ is integrable on $[a,b]$ by Riemann's Criterion for Integrability, and the Integral Lies between Lower and Upper Sums (i).

Proof of (ii). Let $u,v\in[a,b]$ with $u<v$. Then $[u,v]\subseteq[a,b]$ by Basic Facts about Intervals of the Real Line (v) and Basic Facts about Intervals of the Real Line (ii), and the restriction $f|_{[u,v]}:[u,v]\to\mathbb{R}$ is continuous on $[u,v]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vi). By (i), applied to $u<v$ and $f|_{[u,v]}$, this restriction is bounded and integrable on $[u,v]$; by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi) this says that $f$ is integrable on $[u,v]$.