Riemann–Lebesgue lemma for continuous functions on a compact interval

Given: Reals $a<b$, a continuous $f:[a,b]\to \mathbb{R}$, and a real $\epsilon >0$.

For $a\le b$, every continuous real function on $[a,b]$ is a uniform limit of polynomials (Polynomials are uniformly dense in $C([a,b],\mathbb{R})$ for every closed interval).

If integrable functions $u,v$ satisfy $\mathrm{\mid }u(x)-v(x)\mathrm{\mid }\le \eta$ between endpoints, then $\mathrm{\mid }\int u-\int v\mathrm{\mid }\le \eta$ times the endpoint distance (Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error).

If $u,v$ are differentiable on $[a,b]$ with integrable derivatives, then ${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v$ (If $u,v$ are differentiable on $[a,b]$ with $u^{\mathrm{\prime }},v^{\mathrm{\prime }}$ integrable, then ${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v$).

The derivative formulas for sine and cosine, together with the chain rule, give $(\sin nt{)}^{\mathrm{\prime }}=n\cos nt$ and $(\cos nt{)}^{\mathrm{\prime }}=-n\sin nt$ for positive integers $n$ (The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Polynomials and their derivatives exist by the power and derivative-algebra rules; polynomials are continuous, differentiable functions are continuous, and continuous functions on $[a,b]$ are integrable (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, A function differentiable at $c$ is continuous at $c$, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

Absolute values and products of integrable functions are integrable (If $f,g$ are integrable on $[a,b]$ then so are $\mid f\mid$, $f^2$, $fg$, $\max (f,g)$ and $\min (f,g)$, and $\mid {\int }_a^{b}f\mid \le {\int }_a^b\mid f\mid$, claim 1).

The integral is linear and monotone, and a constant $c$ has integral $c(b-a)$ (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$, If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$).

For every real $\eta >0$ there is a positive integer $N$ with $1\mathrm{/}N<\eta$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

For every real $x$, $\mathrm{\mid }\sin x\mathrm{\mid }\le 1$ and $\mathrm{\mid }\cos x\mathrm{\mid }\le 1$ (Parity and the Pythagorean identity for sine and cosine).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two).

If $h$ is integrable on $[a,b]$, then $\mathrm{\mid }{\int }_a^{b}h\mathrm{\mid }\le {\int }_a^b\mathrm{\mid }h\mathrm{\mid }$ (If $f,g$ are integrable on $[a,b]$ then so are $\mid f\mid$, $f^2$, $fg$, $\max (f,g)$ and $\min (f,g)$, and $\mid {\int }_a^{b}f\mid \le {\int }_a^b\mid f\mid$, claim 3).

A real sequence converges to zero when, for every positive rational $\epsilon$, its terms are eventually smaller than $\epsilon$ in absolute value (Limits and Cauchy sequences of reals).