Proof of Term-by-Term Differentiation of a Series of Continuously Differentiable Functions with Summable Uniform Bounds

Each result cited is universally quantified over the data in its own statement. Throughout, $x\in\mathbb{R}^{q}$ and $i\in[q]$ are fixed, $e_{i}$ is the $i$th standard basis vector of $\mathbb{R}^{q}$, and $x+te_{i}$ is the point whose $i$th coordinate is $x_{i}+t$ and whose other coordinates are those of $x$ (Euclidean Points as Tuples of Real Numbers); for $\phi:\mathbb{R}^{q}\to\mathbb{R}$ we write $\phi^{x,i}:[-1,1]\to\mathbb{R}$, $\phi^{x,i}(t)=\phi(x+te_{i})$, for the section of $\phi$ along the $i$th coordinate line through $x$, where $[-1,1]$ is the closed interval and every $t$ with $-1<t<1$ is an interior point of it by Basic Facts about Intervals of the Real Line and Their Interior Points §closed-interval. The map $t\mapsto x+te_{i}$ from $[-1,1]$ to $\mathbb{R}^{q}$ satisfies $d_{E}(x+te_{i},x+t'e_{i})=|t-t'|$ (claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the difference of the two points having the single nonzero coordinate $t-t'$), hence is continuous; so the section of a continuous $\phi$ is continuous on $[-1,1]$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. Moreover, if the partial derivative $\partial_{i}\phi(x+t_{0}e_{i})$ exists for some $-1<t_{0}<1$, then $\phi^{x,i}$ is differentiable at $t_{0}$ with derivative $\partial_{i}\phi(x+t_{0}e_{i})$: the defining condition of Partial Derivative on a Euclidean Open Set at the point $x+t_{0}e_{i}$, in which the point $(a_{1},\dots,a_{i}+h,\dots,a_{q})$ is $x+(t_{0}+h)e_{i}$, is the defining condition of Derivative at an Interior Point for $\phi^{x,i}$ at $t_{0}$ once $\delta$ is decreased so that $|h|<\delta$ forces $t_{0}+h\in[-1,1]$, which is possible since $t_{0}$ is interior.

Claim 1. For every $k$ one has $|f_{k}(x)|\le a_{k}$ and $|\partial_{i}f_{k}(x)|\le b_{k}$, and the series $\sum a_{k}$ and $\sum b_{k}$ converge; so $\sum_{k}f_{k}(x)$ and $\sum_{k}\partial_{i}f_{k}(x)$ converge absolutely by An Absolutely Convergent Series of Real Numbers Converges §dominated, which also gives $|F(x)|\le\sum_{k}a_{k}$ and $|G_{i}(x)|\le\sum_{k}b_{k}$, the bounds asserted in claim 2.

Claim 2, continuity of the partial sums and of the limits. For $m\in\mathbb{N}$ let $S_{m}=\sum_{k=1}^{m}f_{k}$ and $S^{i}_{m}=\sum_{k=1}^{m}\partial_{i}f_{k}$ be the pointwise finite sums, so that $S_{m}(y)$ and $S_{m}^{i}(y)$ are the partial sums of the two series at $y$. Let $T$ be the set of $m\in\mathbb{N}$ such that $S_{m}$ is of class $C^{1}$ on $\mathbb{R}^{q}$ with $\partial_{i}S_{m}=S^{i}_{m}$ for every $i\in[q]$. Then $1\in T$ since $S_{1}=f_{1}$ and $S^{i}_{1}=\partial_{i}f_{1}$ (claim 1 of Properties of Finite Sums); and if $m\in T$ then $S_{m+1}=S_{m}+f_{m+1}$ (claim 1 of Properties of Finite Sums) is of class $C^{1}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, with $\partial_{i}S_{m+1}=\partial_{i}S_{m}+\partial_{i}f_{m+1}=S^{i}_{m+1}$ by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and again claim 1 of Properties of Finite Sums, so $m+1\in T$. By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$. In particular every $S_{m}$ and every $S^{i}_{m}$ is continuous on $\mathbb{R}^{q}$ (a function of class $C^{1}$ is continuous, and $\partial_{i}S_{m}$ is continuous, by clause 1 of C^k Maps on a Euclidean Open Set read through its clause 3). By The Weierstrass M-Test §uniform, applied with $S=\mathbb{R}^{q}$ and the bounds $a_{k}$, respectively $b_{k}$, the partial sums $S_{m}$ converge to $F$ uniformly on $\mathbb{R}^{q}$ and the partial sums $S^{i}_{m}$ converge to $G_{i}$ uniformly on $\mathbb{R}^{q}$; hence $F$ and every $G_{i}$ are continuous on $\mathbb{R}^{q}$ by Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Functions §global.

Claim 2, the partial derivative of $F$. Put $\phi_{k}=f_{k}^{x,i}$ and $h_{k}=(\partial_{i}f_{k})^{x,i}$, continuous on $[-1,1]$ by the preamble, and $G=G_{i}^{x,i}$, continuous on $[-1,1]$ by the previous paragraph. By the preamble, $\phi_{k}$ is differentiable at every $t$ with $-1<t<1$ with derivative $h_{k}(t)$. Let $\Gamma_{k}:[-1,1]\to\mathbb{R}$, $\Gamma_{k}(t)=\int_{-1}^{t}h_{k}(\tau)\,d\tau$, and $\Gamma:[-1,1]\to\mathbb{R}$, $\Gamma(t)=\int_{-1}^{t}G(\tau)\,d\tau$, be the indefinite Riemann integrals of claim 1 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval; by claims 2 and 3 of that theorem $\Gamma_{k}$ and $\Gamma$ are continuous on $[-1,1]$ and differentiable at every $t$ with $-1<t<1$, with $\Gamma_{k}'(t)=h_{k}(t)$ and $\Gamma'(t)=G(t)$.

Step (a): $\phi_{k}(t)-\phi_{k}(0)=\Gamma_{k}(t)-\Gamma_{k}(0)$ for every $t\in[-1,1]$. The function $\Gamma_{k}-\phi_{k}$ is continuous on $[-1,1]$ (claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied to the sum of $\Gamma_{k}$ and the multiple $(-1)\phi_{k}$) and differentiable at every interior point with derivative $h_{k}(t)-h_{k}(t)=0$ by claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives. Let $u<v$ be points of $[-1,1]$. The restriction of $\Gamma_{k}-\phi_{k}$ to $[u,v]$ is continuous on $[u,v]$ (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map) and differentiable at every $t$ with $u<t<v$ with derivative $0$, the defining condition of Derivative at an Interior Point for the restriction being implied by that for the function; so by Mean Value Theorem on a Closed Real Interval there is $c$ with $u<c<v$ and $(\Gamma_{k}-\phi_{k})(v)-(\Gamma_{k}-\phi_{k})(u)=0\cdot(v-u)=0$ (claim 1 of Zero Products and Elementary Identities in a Field). Applying this with $\{u,v\}=\{0,t\}$ for $t\ne0$ gives step (a); for $t=0$ it is trivial.

Step (b): $\Gamma(t)=\sum_{k=1}^{\infty}\Gamma_{k}(t)$ for every $t\in[-1,1]$. For $t=-1$ both sides are $0$ by the degenerate-interval convention in claim 1 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval. Let $-1<t\le1$. By claim 3 of Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval, applied on $[-1,t]$ to the restrictions of $h_{k}$ and of $G$, which are continuous on $[-1,t]$ (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map), one has $\Gamma_{k}(t)=\int_{\mathbb{R}}\tilde{h}_{k}\,d\lambda$ and $\Gamma(t)=\int_{\mathbb{R}}\tilde{G}\,d\lambda$, where $\tilde{h}_{k}$ and $\tilde{G}$ are the zero extensions off $[-1,t]$ and $\lambda$ is Lebesgue measure, these zero extensions being measurable and integrable by claim 2 there. For $m\in\mathbb{N}$ the finite sum $\sum_{k=1}^{m}\tilde{h}_{k}$ is the zero extension of the restriction of $S^{i}_{m}$'s section $(S^{i}_{m})^{x,i}=\sum_{k=1}^{m}h_{k}$ to $[-1,t]$; it converges pointwise on $\mathbb{R}$ to $\tilde{G}$, since $S^{i}_{m}\to G_{i}$ pointwise (claim 1 of Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Functions with $T=S=\mathbb{R}^{q}$) and both sides vanish off $[-1,t]$; and it is dominated by the function $b\,\mathbf{1}_{[-1,t]}$, $b=\sum_{k}b_{k}$, since $|\sum_{k=1}^{m}h_{k}(\tau)|\le\sum_{k=1}^{m}|h_{k}(\tau)|\le\sum_{k=1}^{m}b_{k}\le b$ by claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion; the finite sums $\sum_{k=1}^{m}\tilde{h}_{k}$ are measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. The dominating function is measurable (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and integrable: it is a nonnegative multiple of the indicator of the Borel set $[-1,t]$, whose integral is $b\,\lambda([-1,t])=b(t+1)<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, the integral of an indicator being the measure of the set (Simple Function and Its Integral with Lebesgue Integral of a Nonnegative Measurable Function) and $\lambda([-1,t])=t+1$ by claim 4 of Existence of Lebesgue Measure on the Real Line. By Dominated Convergence Theorem, $\int_{\mathbb{R}}\sum_{k=1}^{m}\tilde{h}_{k}\,d\lambda\to\int_{\mathbb{R}}\tilde{G}\,d\lambda=\Gamma(t)$; and by claim 2 of Linearity and Monotonicity of the Lebesgue Integral (induction on $m$ as in the previous paragraph) the left side equals $\sum_{k=1}^{m}\Gamma_{k}(t)$, the $m$th partial sum of $\sum_{k}\Gamma_{k}(t)$. Hence that series converges with sum $\Gamma(t)$, by Series of Real Numbers §convergent.

Step (c): conclusion. For $t\in[-1,1]$, by Series of Real Numbers §convergent and Elementary Properties of Series of Real Numbers §linearity, step (a) and step (b),

Thus $F^{x,i}(t)=\Gamma(t)+\bigl(F(x)-\Gamma(0)\bigr)$ on $[-1,1]$, and by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (sum with a constant function, whose derivative is $0$, at the interior point $0$) $F^{x,i}$ is differentiable at $0$ with derivative $\Gamma'(0)=G(0)=G_{i}(x)$. By the identification in the preamble, read in the converse direction (the defining condition of Derivative at an Interior Point for $F^{x,i}$ at $0$ with $\delta\le1$ is the defining condition of Partial Derivative on a Euclidean Open Set at $x$, the point $x+he_{i}$ lying in the open set $\mathbb{R}^{q}$), the partial derivative $\partial_{i}F(x)$ exists and equals $G_{i}(x)$. As $x$ and $i$ were arbitrary, $\partial_{i}F=G_{i}$ on $\mathbb{R}^{q}$, and $F$ is continuous with continuous partial derivatives, hence of class $C^{1}$ on $\mathbb{R}^{q}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set. The bounds were obtained in claim 1.