Proof of Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine

Each result cited below is universally quantified over the data in its own statement; it is applied to the data named at the point of use. Write $S$ for the successor map of the natural numbers, so that $2k+1=S(2k)$ for $k\in\mathbb{N}$ by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i), and write $\mathrm{id}$ for the function from $\mathbb{R}$ to $\mathbb{R}$ with $\mathrm{id}(x)=x$. The families $(c_{k})_{k\in\mathbb{N}}$ and $(\sigma_{k})_{k\in\mathbb{N}}$ of the statement are sequences in the set of functions from $\mathbb{R}$ to $\mathbb{R}$. Write $\gamma,\eta:\mathbb{R}\to\mathbb{R}$ for the two functions named in clause 1 of the statement, $\gamma(x)=\cos x-1$ and $\eta(x)=\sin x-x$, and $(p_{m})_{m\in\mathbb{N}}$, $(q_{m})_{m\in\mathbb{N}}$ for the partial sums of $(c_{k})_{k\in\mathbb{N}}$ and of $(\sigma_{k})_{k\in\mathbb{N}}$, so that $p_{m}(x)=\sum_{k=1}^{m}c_{k}(x)$ and $q_{m}(x)=\sum_{k=1}^{m}\sigma_{k}(x)$; by The Real Sine and Cosine Functions (i) and The Real Sine and Cosine Functions (ii) the series $\sum_{k=1}^{\infty}c_{k}(x)$ and $\sum_{k=1}^{\infty}\sigma_{k}(x)$ converge, with sums $\gamma(x)$ and $\eta(x)$, for every $x\in\mathbb{R}$. Every subset of $\mathbb{R}$ carries the metric $d_{\mathbb{R}}$, and every point of the interval $\mathbb{R}$ is an interior point of it by Basic Facts about Intervals of the Real Line (i). Constant multiples, sums and products of functions are formed pointwise, as in the statements of Continuity of Sums and Products of Real-Valued Functions on a Metric Space and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers. Those two items state their hypotheses for a pair of functions; in every application below in which only one function is named, the second function required by their hypotheses is taken to be that same function, so that the hypotheses are met and the conclusions drawn about a constant multiple, and about a constant function, are available. Natural numbers occur below both as exponents and as elements of $\mathbb{R}$; in the latter role they are always read through the canonical map. Finally, $2$ denotes the natural number $1+1$.

Claim 0 (a cancellation identity). Let $v,w$ be nonzero real numbers. Then $vw\neq0$ and $w(vw)^{-1}=v^{-1}$.

That $vw\neq0$ is claim 3 of Zero Products and Elementary Identities in a Field, read contrapositively. Using commutativity and associativity of multiplication in the field $\mathbb{R}$,

$$\bigl(w(vw)^{-1}\bigr)(vw)=w\bigl((vw)^{-1}(vw)\bigr)=w, \qquad v^{-1}(vw)=(v^{-1}v)w=w .$$

Hence $\bigl(w(vw)^{-1}\bigr)(vw)=v^{-1}(vw)$, and multiplying both sides by $(vw)^{-1}$ gives $w(vw)^{-1}=v^{-1}$.

Claim 1 (absolute values of the terms). For every $k\in\mathbb{N}$ and every $x\in\mathbb{R}$,

$$|c_{k}(x)|=\frac{|x|^{2k}}{(2k)!}, \qquad |\sigma_{k}(x)|=\frac{|x|^{2k+1}}{(2k+1)!}.$$

By claim 1 of Elementary Arithmetic in an Ordered Field one has $0\le1$, so $|1|=1$ by Absolute Value in an Ordered Field, and claim 2 of Properties of the Absolute Value in an Ordered Field gives $|-1|=1$; hence $|(-1)^{k}|=|-1|^{k}=1^{k}=1$, using $|c^{n}|=|c|^{n}$ from Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i) and claim 2 of Properties of Natural Number Powers in a Field. The factorials $(2k)!$ and $(2k+1)!$ are positive, hence nonzero and invertible, by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (ii); for a positive real $w$ one has $|w|=w$ by Absolute Value in an Ordered Field, and claim 4 of Properties of the Absolute Value in an Ordered Field gives $|w|\,|w^{-1}|=|w\,w^{-1}|=|1|=1$, whence $|w^{-1}|=w^{-1}$. Applying claim 4 of Properties of the Absolute Value in an Ordered Field twice more, together with $|x^{n}|=|x|^{n}$, yields the two identities.

Claim 2 (clause 1 of the statement). Let $R$ be a positive real number.

Since $0<R$, claim 4 of Elementary Order Arithmetic in an Ordered Field gives $-R<0$, and $0\le R$, so claim 2 of that lemma gives $-R<R$; thus $[-R,R]$ is the closed interval determined by $-R$ and $R$, and it is an interval by Basic Facts about Intervals of the Real Line (v).

Let $x\in[-R,R]$, that is $-R\le x\le R$. By claim 6 of Properties of the Absolute Value in an Ordered Field this gives $|x|\le R$, and $0\le|x|$ by claim 1 of that lemma. Hence claim 5 of Properties of Natural Number Powers in a Field gives $|x|^{n}\le R^{n}$ for every $n\in\mathbb{N}$. The inverse of the positive number $(2k)!$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, hence nonnegative, so claim 5 of Elementary Arithmetic in an Ordered Field gives

$$|x|^{2k}\bigl((2k)!\bigr)^{-1}\le R^{2k}\bigl((2k)!\bigr)^{-1},$$

which with Claim 1 gives $|c_{k}(x)|\le R^{2k}/(2k)!$; in the same way $|\sigma_{k}(x)|\le R^{2k+1}/(2k+1)!$.

Put $M_{k}=R^{2k}/(2k)!$ and $N_{k}=R^{2k+1}/(2k+1)!$ for $k\in\mathbb{N}$. Since $0<R$ gives $0\le R$, Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (iv), applied with $A=R$, gives $0\le M_{k}$ and $0\le N_{k}$ for every $k$, and the convergence of $\sum_{k=1}^{\infty}M_{k}$ and of $\sum_{k=1}^{\infty}N_{k}$; these are the two infinite series appearing in the tail bounds of clause 1.

Now apply The Weierstrass M-Test with $D=\mathbb{R}$, $S=[-R,R]$, the sequence $(c_{k})_{k\in\mathbb{N}}$, the majorant sequence $(M_{k})_{k\in\mathbb{N}}$ and the function $f=\gamma$. Its hypotheses hold: $0\le M_{k}$, the series $\sum_{k=1}^{\infty}M_{k}$ converges, $|c_{k}(x)|\le M_{k}$ for every $k\in\mathbb{N}$ and every $x\in[-R,R]$ as just shown, and $\gamma(x)=\cos x-1=\sum_{k=1}^{\infty}c_{k}(x)$ for every $x\in\mathbb{R}$ by The Real Sine and Cosine Functions (i), hence in particular for every $x\in[-R,R]$. Clause The Weierstrass M-Test (iii) gives that $\sum_{k=1}^{\infty}c_{k}$ converges uniformly to $\gamma$ on $[-R,R]$, and clause The Weierstrass M-Test (ii) gives

$$|\gamma(x)-p_{m}(x)|\le\sum_{k=1}^{\infty}M_{k}-\sum_{k=1}^{m}M_{k}$$

for every $m\in\mathbb{N}$ and every $x\in[-R,R]$; since $\gamma(x)=\cos x-1$, this is the first tail bound. The same argument with the sequence $(\sigma_{k})_{k\in\mathbb{N}}$, the majorants $(N_{k})_{k\in\mathbb{N}}$ and the function $f=\eta$, using The Real Sine and Cosine Functions (ii), gives the remaining assertions of clause 1.

Claim 3 (the terms and the partial sums are continuous on $\mathbb{R}$). Let $k\in\mathbb{N}$. By Continuity of the Identity Map, of Powers, and of Polynomial Functions on a Subset of the Real Line (ii), applied with $E=\mathbb{R}$ and the exponent $2k$, the map $x\mapsto x^{2k}$ is continuous on $\mathbb{R}$. The function $c_{k}$ is the constant multiple of that map by the real number $(-1)^{k}\bigl((2k)!\bigr)^{-1}$, hence continuous on $\mathbb{R}$ by claims 4 and 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied with $(X,d)=(\mathbb{R},d_{\mathbb{R}})$, $A=\mathbb{R}$ and both of the functions named there taken to be that power map. The same argument with the exponent $2k+1$ shows that $\sigma_{k}$ is continuous on $\mathbb{R}$.

By Series of Real-Valued Functions and Their Partial Sums (i), $p_{m}(x)=\sum_{k=1}^{m}c_{k}(x)$ for $x\in\mathbb{R}$. Claim 1 of Properties of Finite Sums gives $\sum_{k=1}^{1}a_{k}=a_{1}$ and $\sum_{k=1}^{S(m)}a_{k}=\bigl(\sum_{k=1}^{m}a_{k}\bigr)+a_{S(m)}$, so $p_{1}=c_{1}$ and $p_{S(m)}=p_{m}+c_{S(m)}$ as functions on $\mathbb{R}$. Since $c_{1}$ is continuous on $\mathbb{R}$, and since claims 2 and 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space show that $p_{m}+c_{S(m)}$ is continuous on $\mathbb{R}$ whenever $p_{m}$ is, the principle of induction, applied to the set of those $m\in\mathbb{N}$ for which $p_{m}$ is continuous on $\mathbb{R}$, gives that $p_{m}$ is continuous on $\mathbb{R}$ for every $m\in\mathbb{N}$. The same argument applies to $(q_{m})_{m\in\mathbb{N}}$.

Claim 4 (clause 2 of the statement). Let $x_{0}\in\mathbb{R}$ and put $R=1+|x_{0}|$.

By claim 1 of Properties of the Absolute Value in an Ordered Field, $0\le|x_{0}|$, and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field; claim 3 of that lemma, applied with $0<1$ and $0\le|x_{0}|$, gives $0=0+0<1+|x_{0}|=R$, so $R$ is positive.

The ball of radius $1$ about $x_{0}$ lies in $[-R,R]$. Let $y$ belong to the open ball $B_{d_{\mathbb{R}}}(x_{0},1)$, that is $|y-x_{0}|<1$. By claim 5 of Properties of the Absolute Value in an Ordered Field, $|y|=|(y-x_{0})+x_{0}|\le|y-x_{0}|+|x_{0}|$. Since $|y-x_{0}|<1$ we have $|y-x_{0}|\le1$, hence $0\le1-|y-x_{0}|$ by claim 3 of Elementary Arithmetic in an Ordered Field; as

$$\bigl(1+|x_{0}|\bigr)-\bigl(|y-x_{0}|+|x_{0}|\bigr)=1-|y-x_{0}|,$$

that same claim gives $|y-x_{0}|+|x_{0}|\le R$. Writing $u=\bigl(|y-x_{0}|+|x_{0}|\bigr)-|y|$ and $v=R-\bigl(|y-x_{0}|+|x_{0}|\bigr)$, both nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field, claim 2 of that lemma gives $0\le u+v=R-|y|$, so $|y|\le R$ by claim 3 again. By claim 6 of Properties of the Absolute Value in an Ordered Field this means $-R\le y\le R$, that is $y\in[-R,R]$. Thus $[-R,R]$ contains every point of $B_{d_{\mathbb{R}}}(x_{0},1)$ that lies in $\mathbb{R}$.

Continuity. By Claim 2, applied with this $R$, the series $\sum_{k=1}^{\infty}c_{k}$ converges uniformly to $\gamma$ on $[-R,R]$, which by Series of Real-Valued Functions and Their Partial Sums (iii) says that $(p_{m})_{m\in\mathbb{N}}$ converges uniformly to $\gamma$ on $[-R,R]$. By Claim 3 each $p_{m}$ is continuous at $x_{0}$ relative to $\mathbb{R}$. Apply Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (ii) with $(X,d_{X})=(\mathbb{R},d_{\mathbb{R}})$, $A=\mathbb{R}$, the sequence $(p_{m})_{m\in\mathbb{N}}$, the function $\gamma$, the point $x_{0}$, the radius $r=1$ and the set $[-R,R]$: it gives that $\gamma$ is continuous at $x_{0}$ relative to $\mathbb{R}$. In the same way, using the sequence $(q_{m})_{m\in\mathbb{N}}$ and the function $\eta$, the function $\eta$ is continuous at $x_{0}$ relative to $\mathbb{R}$.

Since $\cos=\gamma+b$, where $b$ is the function on $\mathbb{R}$ with constant value $1$, claims 1 and 2 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space show that $\cos$ is continuous at $x_{0}$ relative to $\mathbb{R}$. Since $\sin=\eta+\mathrm{id}$ and $\mathrm{id}$ is continuous on $\mathbb{R}$ by Continuity of the Identity Map, of Powers, and of Polynomial Functions on a Subset of the Real Line (i), claim 2 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space shows that $\sin$ is continuous at $x_{0}$ relative to $\mathbb{R}$. As $x_{0}\in\mathbb{R}$ was arbitrary, $\gamma$, $\eta$, $\cos$ and $\sin$ are continuous on $\mathbb{R}$, which is clause 2.

Claim 5 (clause 3 of the statement). By claim 4 of Properties of Natural Number Powers in a Field, $0^{n}=0$ for every $n\in\mathbb{N}$, so claim 1 of Zero Products and Elementary Identities in a Field gives $c_{k}(0)=(-1)^{k}\cdot0\cdot\bigl((2k)!\bigr)^{-1}=0$ and, in the same way, $\sigma_{k}(0)=0$, for every $k\in\mathbb{N}$.

By claim 1 of Properties of Finite Sums, $p_{1}(0)=c_{1}(0)=0$ and $p_{S(m)}(0)=p_{m}(0)+c_{S(m)}(0)=p_{m}(0)+0$, so the principle of induction, applied to the set of those $m\in\mathbb{N}$ with $p_{m}(0)=0$, gives $p_{m}(0)=0$ for every $m\in\mathbb{N}$. The sequence $(p_{m}(0))_{m\in\mathbb{N}}$ is therefore constant with value $0$ and converges to $0$ by Constant Sequences and Index-Shifted Sequences of Real Numbers (i); by Series of Real Numbers (ii) this says $\sum_{k=1}^{\infty}c_{k}(0)=0$. Hence $\cos0=1+0=1$ by The Real Sine and Cosine Functions (i). The same argument with $(q_{m})_{m\in\mathbb{N}}$ gives $\sum_{k=1}^{\infty}\sigma_{k}(0)=0$ and, by The Real Sine and Cosine Functions (ii), $\sin0=0+0=0$.

Claim 6 (clause 4 of the statement). Let $x\in\mathbb{R}$ and $k\in\mathbb{N}$. By Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i), $(-x)^{2k}=x^{2k}$ and $(-x)^{2k+1}=-x^{2k+1}$; hence $c_{k}(-x)=c_{k}(x)$, and claim 2 of Zero Products and Elementary Identities in a Field gives $\sigma_{k}(-x)=(-1)\sigma_{k}(x)$.

The sequences $(c_{k}(-x))_{k\in\mathbb{N}}$ and $(c_{k}(x))_{k\in\mathbb{N}}$ are equal, so the series $\sum_{k=1}^{\infty}c_{k}(-x)$ and $\sum_{k=1}^{\infty}c_{k}(x)$ have the same sum, and The Real Sine and Cosine Functions (i) gives

$$\cos(-x)=1+\sum_{k=1}^{\infty}c_{k}(-x)=1+\sum_{k=1}^{\infty}c_{k}(x)=\cos x .$$

The series $\sum_{k=1}^{\infty}\sigma_{k}(x)$ converges by The Real Sine and Cosine Functions (ii). Apply Elementary Properties of Series of Real Numbers (i) with $\lambda=-1$ to the sequence $(\sigma_{k}(x))_{k\in\mathbb{N}}$, taking the companion sequence named in that clause to be $(\sigma_{k}(x))_{k\in\mathbb{N}}$ itself, whose series converges: the series $\sum_{k=1}^{\infty}(-1)\sigma_{k}(x)$ converges with sum $-\sum_{k=1}^{\infty}\sigma_{k}(x)$. Hence

$$\sin(-x)=(-x)+\sum_{k=1}^{\infty}\sigma_{k}(-x)=-x-\sum_{k=1}^{\infty}\sigma_{k}(x)=-\Bigl(x+\sum_{k=1}^{\infty}\sigma_{k}(x)\Bigr)=-\sin x .$$

Claim 7 (antiderivatives of the terms). Let $k\in\mathbb{N}$ and $t_{0}\in\mathbb{R}$, and write $-c_{k+1}$ for the constant multiple of $c_{k+1}$ by $-1$. Then $\sigma_{k}$ is differentiable at $t_{0}$ with derivative $c_{k}(t_{0})$, and $-c_{k+1}$ is differentiable at $t_{0}$ with derivative $\sigma_{k}(t_{0})$.

First assertion. Since $2k+1=S(2k)$, claim 1 of Derivative of a Polynomial Function on the Real Line, applied with $I=\mathbb{R}$ and $m=2k$, gives that the map $t\mapsto t^{2k+1}$ is differentiable at $t_{0}$ with derivative $(2k+1)\,t_{0}^{2k}$, the factor $2k+1$ being read in $\mathbb{R}$ through the canonical map. The factorial $(2k+1)!$ is positive, hence nonzero, by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (ii), and $\sigma_{k}$ is the constant multiple of that map by $(-1)^{k}\bigl((2k+1)!\bigr)^{-1}$; so claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, applied on the interval $\mathbb{R}$ at its interior point $t_{0}$ with both of the functions named there taken to be that power map, gives that $\sigma_{k}$ is differentiable at $t_{0}$ with derivative

$$(-1)^{k}\bigl((2k+1)!\bigr)^{-1}(2k+1)\,t_{0}^{2k}.$$

By Recursion for the Factorial of a Natural Number (i), applied with $n=2k$, one has $(2k+1)!=(2k+1)\cdot(2k)!$; the canonical image of $2k+1$ is positive, hence nonzero, by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $(2k)!$ is nonzero. Since multiplication in $\mathbb{R}$ is commutative, $(2k+1)!=(2k)!\,(2k+1)$, so Claim 0, applied with $w=2k+1$ and $v=(2k)!$, gives $(2k+1)\bigl((2k+1)!\bigr)^{-1}=\bigl((2k)!\bigr)^{-1}$, so the displayed derivative equals $c_{k}(t_{0})$.

Second assertion. Write $j=2k+1$. We first check that $2(k+1)=j+1$. By Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i), applied to the natural number $k+1$, one has $2(k+1)=(k+1)+(k+1)$; by claims 3 and 4 of Arithmetic of Addition on the Natural Numbers, addition on $\mathbb{N}$ being associative and commutative, $(k+1)+(k+1)=(k+k)+(1+1)$; and $k+k=2k$ by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i). Hence, associating once more,

$$2(k+1)=2k+(1+1)=(2k+1)+1=j+1 .$$

By claim 1 of Arithmetic of Addition on the Natural Numbers, $k+1=S(k)$ and $j+1=S(j)$. Hence claim 1 of Properties of Natural Number Powers in a Field gives $(-1)^{k+1}=(-1)^{k}(-1)$, so claim 2 of Zero Products and Elementary Identities in a Field gives

$$-c_{k+1}(t)=-\frac{(-1)^{k}(-1)\,t^{\,j+1}}{(j+1)!}=\frac{(-1)^{k}\,t^{\,j+1}}{(j+1)!} \qquad(t\in\mathbb{R}).$$

Claim 1 of Derivative of a Polynomial Function on the Real Line, applied with $I=\mathbb{R}$ and $m=j$, gives that $t\mapsto t^{\,j+1}$ is differentiable at $t_{0}$ with derivative $(j+1)\,t_{0}^{\,j}$, the factor $j+1$ again read in $\mathbb{R}$ through the canonical map; so claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, with both of the functions named there taken to be that power map, gives that $-c_{k+1}$ is differentiable at $t_{0}$ with derivative $(-1)^{k}\bigl((j+1)!\bigr)^{-1}(j+1)\,t_{0}^{\,j}$. By Recursion for the Factorial of a Natural Number (i), applied with $n=j$, one has $(j+1)!=(j+1)\cdot j!$, which by commutativity is $j!\,(j+1)$; both factors are nonzero, the first by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (ii) and the second by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so Claim 0 with $w=j+1$ and $v=j!$ gives $(j+1)\bigl((j+1)!\bigr)^{-1}=(j!)^{-1}$. Since $j=2k+1$, the derivative equals $(-1)^{k}t_{0}^{2k+1}/(2k+1)!=\sigma_{k}(t_{0})$.

Claim 8 (termwise integration). Let $R$ be a positive real number, let $J=[-R,R]$, and let $u\in J$ with $-R<u$. Write $\int_{-R}^{u}h(t)\,dt$ for the Riemann integral on $[-R,u]$ of the restriction of a function $h:\mathbb{R}\to\mathbb{R}$ to $[-R,u]$, which exists whenever that restriction is continuous on $[-R,u]$, by A Continuous Function on a Closed Interval is Riemann Integrable (i). Then

$$\int_{-R}^{u}\gamma(t)\,dt=\eta(u)-\eta(-R), \qquad \int_{-R}^{u}\eta(t)\,dt=\bigl(c_{1}(u)-\gamma(u)\bigr)-\bigl(c_{1}(-R)-\gamma(-R)\bigr).$$

Put $K=[-R,u]$. Since $-R,u\in J$ and $-R\le u$, Basic Facts about Intervals of the Real Line (ii) gives $K\subseteq J$, and by Basic Facts about Intervals of the Real Line (v) $K$ is an interval every point of which strictly between $-R$ and $u$ is an interior point of $K$. By Claims 3 and 4 and claim 1 of Restriction Stability of Continuity and of the Derivative, the restrictions to $K$ of $c_{k}$, $\sigma_{k}$, $\gamma$ and $\eta$ are continuous on $K$.

The first identity. By Claim 2, $(p_{m})_{m\in\mathbb{N}}$ converges uniformly to $\gamma$ on $J$, so Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (i), applied with $A=\mathbb{R}$, $S=J$ and $T=K$, gives that it converges uniformly to $\gamma$ on $K$. The defining condition of Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii) constrains only the values of the functions at points of $K$, and those values are unchanged by restriction to $K$; moreover the $m$-th partial sum of $(c_{k}|_{K})_{k\in\mathbb{N}}$ takes at $x\in K$ the value $\sum_{k=1}^{m}c_{k}(x)=p_{m}(x)$, so it is $p_{m}|_{K}$. Hence the series $\sum_{k=1}^{\infty}c_{k}|_{K}$ converges uniformly to $\gamma|_{K}$ on $K$ in the sense of Series of Real-Valued Functions and Their Partial Sums (iii), read with $D=S=K$.

Therefore Integrals of Uniform Limits and of Uniformly Convergent Series (iv), applied with $[a,b]=K$, the sequence $(c_{k}|_{K})_{k\in\mathbb{N}}$ and the function $\gamma|_{K}$, gives that $\sum_{k=1}^{\infty}\int_{-R}^{u}c_{k}(t)\,dt$ converges with

$$\sum_{k=1}^{\infty}\int_{-R}^{u}c_{k}(t)\,dt=\int_{-R}^{u}\gamma(t)\,dt .$$

By Claim 7 and claim 2 of Restriction Stability of Continuity and of the Derivative, $\sigma_{k}|_{K}$ is differentiable at each point $t$ of $K$ strictly between $-R$ and $u$, with derivative $c_{k}(t)$ there, and $\sigma_{k}|_{K}$ is continuous on $K$; so Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative, applied on $K$ with $f=c_{k}|_{K}$ and $F=\sigma_{k}|_{K}$, gives $\int_{-R}^{u}c_{k}(t)\,dt=\sigma_{k}(u)-\sigma_{k}(-R)$.

The series $\sum_{k=1}^{\infty}\sigma_{k}(u)$ and $\sum_{k=1}^{\infty}\sigma_{k}(-R)$ converge by The Real Sine and Cosine Functions (ii). By Elementary Properties of Series of Real Numbers (i), applied with $\lambda=-1$ to the sequence $(\sigma_{k}(-R))_{k\in\mathbb{N}}$ and with companion sequence $(\sigma_{k}(u))_{k\in\mathbb{N}}$, the series $\sum_{k=1}^{\infty}(-1)\sigma_{k}(-R)$ converges with sum $-\sum_{k=1}^{\infty}\sigma_{k}(-R)$; the additive part of that clause, applied to the sequences $(\sigma_{k}(u))_{k\in\mathbb{N}}$ and $((-1)\sigma_{k}(-R))_{k\in\mathbb{N}}$, then gives

$$\sum_{k=1}^{\infty}\bigl(\sigma_{k}(u)-\sigma_{k}(-R)\bigr)=\sum_{k=1}^{\infty}\sigma_{k}(u)-\sum_{k=1}^{\infty}\sigma_{k}(-R)=\eta(u)-\eta(-R),$$

because $\eta(x)=\sin x-x=\sum_{k=1}^{\infty}\sigma_{k}(x)$ by The Real Sine and Cosine Functions (ii). This proves the first identity.

The second identity. The same argument, with $(\sigma_{k})_{k\in\mathbb{N}}$ in place of $(c_{k})_{k\in\mathbb{N}}$, $(q_{m})_{m\in\mathbb{N}}$ in place of $(p_{m})_{m\in\mathbb{N}}$, $\eta$ in place of $\gamma$, and the antiderivative $-c_{k+1}$ of $\sigma_{k}$ supplied by Claim 7 in place of $\sigma_{k}$, applies here as well. The one hypothesis that is not merely a relabelling is the continuity of the new antiderivative: $c_{k+1}$ is continuous on $\mathbb{R}$ by Claim 3, applied to the natural number $k+1$, hence so is $-c_{k+1}$ by claim 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space with both of the functions named there taken to be $c_{k+1}$, and therefore the restriction of $-c_{k+1}$ to $K$ is continuous on $K$ by claim 1 of Restriction Stability of Continuity and of the Derivative. The argument thus gives that $\sum_{k=1}^{\infty}\bigl(-c_{k+1}(u)+c_{k+1}(-R)\bigr)$ converges with

$$\sum_{k=1}^{\infty}\bigl(-c_{k+1}(u)+c_{k+1}(-R)\bigr)=\int_{-R}^{u}\eta(t)\,dt .$$

Fix $x\in\mathbb{R}$. The series $\sum_{k=1}^{\infty}c_{k}(x)$ converges by The Real Sine and Cosine Functions (i), so Shifting the Index of a Series of Real Numbers (i), applied to the sequence $(c_{k}(x))_{k\in\mathbb{N}}$, gives that $\sum_{k=1}^{\infty}c_{k+1}(x)$ converges with

$$\gamma(x)=\sum_{k=1}^{\infty}c_{k}(x)=c_{1}(x)+\sum_{k=1}^{\infty}c_{k+1}(x), \qquad\text{so}\qquad \sum_{k=1}^{\infty}c_{k+1}(x)=\gamma(x)-c_{1}(x).$$

Applying Elementary Properties of Series of Real Numbers (i) exactly as above, with $\lambda=-1$ on the sequence $(c_{k+1}(u))_{k\in\mathbb{N}}$ and companion $(c_{k+1}(-R))_{k\in\mathbb{N}}$, gives

$$\int_{-R}^{u}\eta(t)\,dt=-\bigl(\gamma(u)-c_{1}(u)\bigr)+\bigl(\gamma(-R)-c_{1}(-R)\bigr),$$

which is the second identity.

Claim 9 (clause 5 of the statement). Let $x_{0}\in\mathbb{R}$, put $R=1+|x_{0}|$, which is positive by Claim 4, and put $J=[-R,R]$.

Since $0<1$ and $|x_{0}|\le|x_{0}|$, claim 3 of Elementary Order Arithmetic in an Ordered Field gives $0+|x_{0}|<1+|x_{0}|$, that is $|x_{0}|<R$; by claim 9 of Properties of the Absolute Value in an Ordered Field this means $-R<x_{0}<R$, so $x_{0}$ is an interior point of $J$ by Basic Facts about Intervals of the Real Line (v). Moreover $\{y\in\mathbb{R}:|y-x_{0}|<1\}\subseteq J$ by the inclusion established in Claim 4.

The derivative of $\sin$. By Claim 4 and claim 1 of Restriction Stability of Continuity and of the Derivative, $\gamma|_{J}$ is continuous on $J$, so Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity, applied on $J$ with $f=\gamma|_{J}$, defines $G:J\to\mathbb{R}$ by $G(u)=\int_{-R}^{u}\gamma(t)\,dt$ for $u\in J$, with $G(-R)=0$ by its clause 1, and gives by its clause 3 that $G$ is differentiable at $x_{0}$ with derivative $\gamma(x_{0})$.

By Claim 8, $G(u)=\eta(u)-\eta(-R)$ for every $u\in J$ with $-R<u$; the same identity holds at $u=-R$, both sides being $0$. Hence $\eta|_{J}=G+e$, where $e$ is the function on $J$ with constant value $\eta(-R)$. By claims 1 and 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, applied on the interval $J$ at its interior point $x_{0}$, the function $\eta|_{J}$ is differentiable at $x_{0}$ with derivative $\gamma(x_{0})+0=\gamma(x_{0})$.

By claim 1 of Properties of Natural Number Powers in a Field, $x^{1}=x$ for every $x\in\mathbb{R}$, so $\mathrm{id}|_{J}$ is the restriction to $J$ of the map $x\mapsto x^{1}$ and is therefore differentiable at $x_{0}$ with derivative $1$, by claim 1 of Derivative of a Polynomial Function on the Real Line applied with $I=J$. Since $\sin|_{J}=\eta|_{J}+\mathrm{id}|_{J}$, claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers gives that $\sin|_{J}$ is differentiable at $x_{0}$ with derivative $\gamma(x_{0})+1=\cos x_{0}$.

Finally apply Differentiability at an Interior Point is a Local Property (i), taking the interval called $I$ there to be $J$, the interval called $J$ there to be $\mathbb{R}$, the function to be $\sin$, and the radius to be $1$: the hypotheses hold because $x_{0}$ is an interior point of $J$ and of $\mathbb{R}$ and because $\{y\in\mathbb{R}:|y-x_{0}|<1\}\subseteq J$. Hence $\sin$ is differentiable at $x_{0}$ with derivative $\cos x_{0}$.

The derivative of $\cos$. By Claim 4 and claim 1 of Restriction Stability of Continuity and of the Derivative, $\eta|_{J}$ is continuous on $J$, so Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity, applied on $J$ with $f=\eta|_{J}$, defines $H:J\to\mathbb{R}$ by $H(u)=\int_{-R}^{u}\eta(t)\,dt$, with $H(-R)=0$, and gives that $H$ is differentiable at $x_{0}$ with derivative $\eta(x_{0})$. By Claim 8, $H(u)=\bigl(c_{1}(u)-\gamma(u)\bigr)-\bigl(c_{1}(-R)-\gamma(-R)\bigr)$ for every $u\in J$ with $-R<u$, and the same identity holds at $u=-R$, both sides being $0$. Rearranging,

$$\gamma|_{J}=c_{1}|_{J}+(-1)H+e',$$

where $e'$ is the function on $J$ with constant value $-\bigl(c_{1}(-R)-\gamma(-R)\bigr)$.

We compute the derivative of $c_{1}|_{J}$ at $x_{0}$. The exponent occurring in $c_{1}$ is the natural number written $2k$ with $k=1$, which by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials (i) is $1\cdot2$; and

$$1\cdot2=1\cdot S(1)=1\cdot1+1=1+1=2,$$

by clauses 3 and 4 of Natural Numbers and claim 1 of Arithmetic of Addition on the Natural Numbers. Moreover $(-1)^{1}=-1$ by claim 1 of Properties of Natural Number Powers in a Field, and Recursion for the Factorial of a Natural Number (i), applied with $n=1$, gives $2!=(1+1)!=(1+1)\cdot1!=2\cdot1=2$, the value $1!=1$ being the other assertion of that lemma and the factor $1+1$ being read in $\mathbb{R}$ through the canonical map. Hence $c_{1}(x)=-x^{2}\cdot2^{-1}$ for $x\in\mathbb{R}$. Claim 1 of Derivative of a Polynomial Function on the Real Line, applied with $I=J$ and $m=1$, gives that the restriction to $J$ of $x\mapsto x^{S(1)}=x^{2}$ is differentiable at $x_{0}$ with derivative $2\,x_{0}^{1}=2x_{0}$, the factor $2$ being read in $\mathbb{R}$ through the canonical map; so claim 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers, with the constant $-2^{-1}$ and with both of the functions named there taken to be that restriction, gives that $c_{1}|_{J}$ is differentiable at $x_{0}$ with derivative $-2^{-1}\cdot2x_{0}=-x_{0}$.

Applying claims 1 and 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers to the displayed decomposition of $\gamma|_{J}$, that function is differentiable at $x_{0}$ with derivative

$$-x_{0}+(-1)\eta(x_{0})+0=-x_{0}-\bigl(\sin x_{0}-x_{0}\bigr)=-\sin x_{0}.$$

Since $\cos|_{J}=\gamma|_{J}+b$ with $b$ the function on $J$ with constant value $1$, claims 1 and 2 of Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers give that $\cos|_{J}$ is differentiable at $x_{0}$ with derivative $-\sin x_{0}$; and Differentiability at an Interior Point is a Local Property (i), applied exactly as above with the function $\cos$, gives that $\cos$ is differentiable at $x_{0}$ with derivative $-\sin x_{0}$.

As $x_{0}\in\mathbb{R}$ was arbitrary, clause 5 follows, and the proof is complete.