Proof of Shifted Power Maps, and Integrating a Power Bound on the Derivative of a Matrix-Valued Map

Computations with real numbers use The Real Numbers as a Type.

Proof of (i). Let $N\in\mathbb{N}_{0}$. For every $r\in[a,b]$, $(r-s)^{N}/N!$ is a real number, as $N!\neq0$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii); so the map $[a,b]\to\mathbb{R}$, $r\mapsto(r-s)^{N}/N!$, exists (a map given by an expression, Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), and it is the only map $[a,b]\to\mathbb{R}$ with these values by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv). By Powers with Exponents in the Natural Numbers with Zero (ii), $(r-s)^{0}=1$, and $0!=1$ by The Factorial as a Map on the Natural Numbers with Zero (ii); so $\psi_{0}(r)=1/0!=1$ for every $r\in[a,b]$, $\psi_{0}$ is the constant map with value $1$, and it is differentiable at every point by Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i). For $N\in\mathbb{N}$, $1\le N$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (ii), so $0^{N}=0$ by Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii) and $\psi_{N}(s)=0$.

Let $N\in\mathbb{N}_{0}$. By Basic Facts about Intervals of the Real Line (i), $\mathbb{R}$ is an interval containing two distinct points, and by Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iii), with $\mathbb{R}$ in place of $I$, the map $P:\mathbb{R}\to\mathbb{R}$, $x\mapsto x^{N+1}$, is differentiable at every $x\in\mathbb{R}$ with derivative $(N+1)x^{N}$. The map $\tau:[a,b]\to\mathbb{R}$, $r\mapsto r-s$, the sum of the identity map of $[a,b]$, read as a map into $\mathbb{R}$, and the constant map with value $-s$, is differentiable at every $r\in[a,b]$ with derivative $1+0=1$, by Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv). By Chain Rule for Real Functions on Intervals (i), with $[a,b]$ and $\mathbb{R}$ in place of $I$ and $J$, $\tau$ in place of $f$ and $P$ in place of $g$, the map $P\circ\tau:r\mapsto(r-s)^{N+1}$ is differentiable at every $r\in[a,b]$ with derivative $(N+1)(r-s)^{N}$. By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), $\psi_{N+1}=\frac{1}{(N+1)!}(P\circ\tau)$ is differentiable at every $r\in[a,b]$ with derivative

$$\psi_{N+1}'(r)=\frac{(N+1)(r-s)^{N}}{(N+1)!}=\frac{(r-s)^{N}}{N!}=\psi_{N}(r),$$

since $(N+1)!=(N+1)\cdot N!$ and $N!\neq0$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii), and $N+1\neq0$ in $\mathbb{R}$: $N+1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (i) and Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (v), so $1\le N+1$ in $\mathbb{R}$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (ii).

Every $N\in\mathbb{N}_{0}$ is $0$ or $m+1$ for some $m\in\mathbb{N}_{0}$, by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity (vii) and Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (v). So every $\psi_{N}$, $N\in\mathbb{N}_{0}$, is differentiable at every point of $[a,b]$, hence differentiable on $[a,b]$, and continuous at every point of $[a,b]$ by The Increment Form of the Derivative, and Differentiability Implies Continuity (ii), hence continuous on $[a,b]$ (Continuous Map Between Metric Spaces (ii)). This proves (i).

Proof of (ii). Let $p,q,c,N$ and $E$ be as there, and let $t\in[a,b]$. As $N+1\in\mathbb{N}$ (Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (i) and Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (v)), $\psi_{N+1}(s)=0$ by (i). Let $R=c\,|t-s|^{N+1}/(N+1)!$; then $R\ge0$ by $c\ge0$, Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (vi) and Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii). If $t=s$, then $\lVert E(t)\rVert=\lVert0_{p\times q}\rVert=0\le R$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iv).

Let $t\neq s$, let $u$ and $v$ be the smaller and the larger of $s$ and $t$, so that $u<v$, $u,v\in[a,b]$ and $s\in\{u,v\}$, and let $\varrho=1$ if $s=u$ and $\varrho=-1$ if $s=v$. Then $|r-s|=\varrho(r-s)$ for every $r\in[u,v]$, so $|r-s|^{N}=\varrho^{N}(r-s)^{N}$ by Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii), and $|r-s|^{N}/N!=\varrho^{N}\psi_{N}(r)$.

By The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (iii), with $[a,b]$ in place of $I$ and $E$ in place of $A$, $\int_{u}^{v}E'$ is defined and equals $E(v)-E(u)$. As $E(s)=0_{p\times q}$, this is $E(t)-0_{p\times q}=E(t)$ if $t=v$ and $0_{p\times q}-E(t)=-E(t)=(-1)E(t)$ if $t=u$ (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii)); in either case $\lVert E(t)\rVert=\bigl\lVert\int_{u}^{v}E'\bigr\rVert$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (v). By Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (viii), with $[a,b]$ in place of $I$ and $E'$ in place of $A$, $\int_{u}^{v}\lVert E'(r)\rVert\,dr$ is defined and

$$\Bigl\lVert\int_{u}^{v}E'\Bigr\rVert\le\int_{u}^{v}\lVert E'(r)\rVert\,dr .$$

The map $r\mapsto\lVert E'(r)\rVert$ is continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), with $[a,b]$ in place of $S$. Let $g=(c\,\varrho^{N})\psi_{N}:[a,b]\to\mathbb{R}$, the scalar multiple; it is continuous on $[a,b]$ by (i) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v). For $r\in[u,v]$, by the hypothesis on $E'$,

$$\lVert E'(r)\rVert\le c\,\frac{|r-s|^{N}}{N!}=c\,\varrho^{N}\psi_{N}(r)=g(r).$$

The maps $r\mapsto\lVert E'(r)\rVert$ and $g$ are continuous on $[a,b]$, so they are integrable on $[u,v]$ by A Continuous Function on a Closed Interval is Riemann Integrable (ii); that is, by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), their restrictions to $[u,v]$ are bounded and integrable on $[u,v]$, with integrals $\int_{u}^{v}\lVert E'(r)\rVert\,dr$ and $\int_{u}^{v}g$. By Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii), with $u$ and $v$ in place of $a$ and $b$ and these restrictions in place of $f$ and $g$, $\int_{u}^{v}\lVert E'(r)\rVert\,dr\le\int_{u}^{v}g$. By Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iv), $\int_{u}^{v}\psi_{N}$ is defined, and by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (ii) and Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (vi), with $[a,b]$ in place of $I$ and $\psi_{N+1}$ in place of $f$, whose derivative is $\psi_{N}$ by (i),

$$\int_{u}^{v}g=c\,\varrho^{N}\int_{u}^{v}\psi_{N}=c\,\varrho^{N}\bigl(\psi_{N+1}(v)-\psi_{N+1}(u)\bigr).$$

If $s=u$, then $\varrho=1$, $\varrho^{N}=1$ by Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii), $t=v$ and $\psi_{N+1}(u)=0$, so the right side is $c\,(t-s)^{N+1}/(N+1)!=R$, as $t-s=|t-s|$. If $s=v$, then $\varrho=-1$, $t=u$ and $\psi_{N+1}(v)=0$; by Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (i) and Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii), $-(-1)^{N}=(-1)^{N}(-1)^{1}=(-1)^{N+1}$ and $(-1)^{N+1}(t-s)^{N+1}=(s-t)^{N+1}$, so the right side is $c\,(s-t)^{N+1}/(N+1)!=R$, as $s-t=|t-s|$. Combining the inequalities, $\lVert E(t)\rVert\le R$. As $t\in[a,b]$ was arbitrary, this proves (ii).