Proof of Linearity of the Riemann Integral of Continuous Functions, and Passage to a Uniform Limit

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named in the step where it is cited. Write $(a,b)$ for $\{x\in\mathbb{R}:a<x<b\}$; as recorded in the preamble of Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative, every $x\in(a,b)$ is an interior point of the interval $[a,b]$. Since $a<b$, claim 3 of Elementary Arithmetic in an Ordered Field gives $0\le b-a$, and $b-a\ne0$, so $b-a$ is positive. Continuity of sums and constant multiples of real-valued functions is taken from clause 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied throughout with the metric space $(\mathbb{R},d_{\mathbb{R}})$ and the subset $[a,b]$.

Claim 1. The functions $h_{1}+h_{2}$ and $c\,h_{1}$ are continuous on $[a,b]$ by clause 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. Hence all four integrals appearing in the claim exist, by A Continuous Function on a Closed Interval is Riemann Integrable (i).

For $i\in\{1,2\}$ let $H_{i}:[a,b]\to\mathbb{R}$ be the indefinite integral $H_{i}(x)=\int_{a}^{x}h_{i}(t)\,dt$ of claim 1 of Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity, applied to the continuous function $h_{i}$. By claims 2 and 3 of that theorem, $H_{i}$ is continuous on $[a,b]$ and differentiable at every $x\in(a,b)$ with $H_{i}'(x)=h_{i}(x)$. By the same claim 1, $H_{i}(a)=0$ and $H_{i}(b)=\int_{a}^{b}h_{i}(t)\,dt$.

The function $H_{1}+H_{2}$ is continuous on $[a,b]$, and for every $x\in(a,b)$ it is differentiable at $x$ with derivative $h_{1}(x)+h_{2}(x)$, by claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers applied with the interval $[a,b]$, the functions $H_{1}$ and $H_{2}$, and the interior point $x$. The function $h_{1}+h_{2}$ is Riemann integrable on $[a,b]$, so Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative, applied with the integrand $h_{1}+h_{2}$ and the function $H_{1}+H_{2}$, gives

$$\int_{a}^{b}\bigl(h_{1}+h_{2}\bigr)(t)\,dt=(H_{1}+H_{2})(b)-(H_{1}+H_{2})(a)=\int_{a}^{b}h_{1}(t)\,dt+\int_{a}^{b}h_{2}(t)\,dt .$$

The same argument with $c\,H_{1}$ in place of $H_{1}+H_{2}$, using the constant-multiple assertion of claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, gives

$$\int_{a}^{b}\bigl(c\,h_{1}\bigr)(t)\,dt=c\,H_{1}(b)-c\,H_{1}(a)=c\int_{a}^{b}h_{1}(t)\,dt .$$

Claim 2. By claim 6 of Properties of the Absolute Value in an Ordered Field, the hypothesis $|h(t)|\le M$ is equivalent to $-M\le h(t)\le M$, for every $t\in[a,b]$. The function $h$ is Riemann integrable on $[a,b]$ by A Continuous Function on a Closed Interval is Riemann Integrable (i), so claim 2 of Uniform Partitions and Order Bounds for the Riemann Integral, applied with $p=a$, $q=b$, the lower bound $-M$ and the upper bound $M$, gives

$$(-M)\,(b-a)\le\int_{a}^{b}h(t)\,dt\le M\,(b-a) .$$

Since $(-M)(b-a)=-\bigl(M\,(b-a)\bigr)$ in the field $\mathbb{R}$, claim 6 of Properties of the Absolute Value in an Ordered Field converts this two-sided bound into the assertion.

Claim 3. That $h$ is continuous on $[a,b]$ is Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (iii), applied with the metric space $(\mathbb{R},d_{\mathbb{R}})$, the subset $A=[a,b]$, the sequence $(h_{k})_{k\in\mathbb{N}}$ and the function $h$. In particular $\int_{a}^{b}h(t)\,dt$ exists.

Let $\varepsilon$ be a real number with $0<\varepsilon$. The choices are made in this order: $\varepsilon$ is given, then $\eta$, then $K$. By claim 8 of Elementary Order Arithmetic in an Ordered Field the number $\varepsilon/2$ is positive and satisfies $\varepsilon/2<\varepsilon$; since $b-a$ is positive, claim 7 of that lemma makes $(b-a)^{-1}$ positive, and claim 5 of that lemma makes

$$\eta=\frac{\varepsilon}{2\,(b-a)}$$

positive. Moreover $\eta\,(b-a)=\varepsilon/2<\varepsilon$.

By Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii) there is $K\in\mathbb{N}$ such that $|h_{k}(t)-h(t)|<\eta$, and hence $|h_{k}(t)-h(t)|\le\eta$, for every $k\in\mathbb{N}$ with $K\le k$ and every $t\in[a,b]$.

Fix such a $k$. The function $h_{k}+(-1)\,h$ is continuous on $[a,b]$ and takes the value $h_{k}(t)-h(t)$ at each $t\in[a,b]$. Applying Claim 1 twice, once with the constant $-1$ and the function $h$ and once with the pair $h_{k}$ and $(-1)h$,

$$\int_{a}^{b}\bigl(h_{k}+(-1)h\bigr)(t)\,dt=\int_{a}^{b}h_{k}(t)\,dt-\int_{a}^{b}h(t)\,dt .$$

Applying Claim 2 to the function $h_{k}+(-1)h$ with the bound $M=\eta$,

$$\Bigl|\int_{a}^{b}h_{k}(t)\,dt-\int_{a}^{b}h(t)\,dt\Bigr|\le\eta\,(b-a)<\varepsilon .$$

As $\varepsilon>0$ was arbitrary, Limit of a Sequence of Real Numbers gives that $\bigl(\int_{a}^{b}h_{k}(t)\,dt\bigr)_{k\in\mathbb{N}}$ converges to $\int_{a}^{b}h(t)\,dt$.

Claim 4. Let $(s_{m})_{m\in\mathbb{N}}$ be the sequence of partial sums of $(g_{k})_{k\in\mathbb{N}}$.

Step 1 (partial sums, by induction). We show by induction on $m\in\mathbb{N}$ that $s_{m}$ is continuous on $[a,b]$ and that

$$\int_{a}^{b}s_{m}(t)\,dt=\sum_{k=1}^{m}\int_{a}^{b}g_{k}(t)\,dt .$$

For $m=1$, claim 1 of Properties of Finite Sums gives $s_{1}(t)=g_{1}(t)$ for every $t\in[a,b]$ and $\sum_{k=1}^{1}\int_{a}^{b}g_{k}(t)\,dt=\int_{a}^{b}g_{1}(t)\,dt$, so both assertions hold. Assume they hold for some $m\in\mathbb{N}$. By claim 1 of Properties of Finite Sums, $s_{m+1}(t)=s_{m}(t)+g_{m+1}(t)$ for every $t\in[a,b]$, that is, $s_{m+1}=s_{m}+g_{m+1}$; this function is continuous on $[a,b]$ by clause 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. By Claim 1, then the induction hypothesis, then claim 1 of Properties of Finite Sums once more,

$$\int_{a}^{b}s_{m+1}(t)\,dt=\int_{a}^{b}s_{m}(t)\,dt+\int_{a}^{b}g_{m+1}(t)\,dt=\sum_{k=1}^{m+1}\int_{a}^{b}g_{k}(t)\,dt .$$

This completes the induction.

Step 2 (conclusion). By hypothesis and Series of Real-Valued Functions and Their Partial Sums (iii), the sequence $(s_{m})_{m\in\mathbb{N}}$ converges uniformly to $g$ on $[a,b]$, and by Step 1 every $s_{m}$ is continuous on $[a,b]$. Claim 3, applied to the sequence $(s_{m})_{m\in\mathbb{N}}$ and the function $g$, shows that $g$ is continuous on $[a,b]$ and that the sequence of real numbers $\bigl(\int_{a}^{b}s_{m}(t)\,dt\bigr)_{m\in\mathbb{N}}$ converges to $\int_{a}^{b}g(t)\,dt$.

By the identity of Step 1, that sequence is the sequence of partial sums of the sequence of real numbers $\bigl(\int_{a}^{b}g_{k}(t)\,dt\bigr)_{k\in\mathbb{N}}$. Hence, by Series of Real Numbers (ii), the series $\sum_{k=1}^{\infty}\int_{a}^{b}g_{k}(t)\,dt$ converges with sum $\int_{a}^{b}g(t)\,dt$.