Trigonometric and Oscillatory Examples in One Variable: Examples and Counterexamples

Given: The parameters $a=1\mathrm{/}2$ and $b=15$.

The unique smallest positive zero $\gamma$ of cosine lies in $(0,2)$ (Cosine has a smallest positive zero, lying strictly between zero and two).

The number $\pi$ is defined by $\pi =2\gamma$ (Pi as twice the smallest positive zero of cosine).

The classical Weierstrass function is $W_{a,b}(x)={\sum }_{n=0}^{\mathrm{\infty }}{a}^n\cos (b^n\pi x)$ (The classical Weierstrass function).

If $0<a<1$, $b>1$ is an odd integer, and $ab>1+3\pi \mathrm{/}2$, then $W_{a,b}$ is continuous everywhere and differentiable nowhere (Under $ab>1+3\pi \mathrm{/}2$, the classical Weierstrass function is continuous everywhere and differentiable nowhere).

For $0<a<1$ and odd integer $b>1$, the defining Weierstrass series converges uniformly on $\mathbb{R}$ (The classical Weierstrass series converges uniformly to a continuous function).

Given: The universal claim in the Statement.

There is a sequence of smooth functions converging uniformly on $\mathbb{R}$ to a continuous function which is differentiable at no real point (A uniform limit of smooth functions need not be differentiable anywhere).

Given: The function $f$ in the Statement.

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

If a function is squeezed near a point between two functions having the same limit there, then it has that limit (If $f\le g\le h$ near $c$ and $f$ and $h$ have the same limit at $c$, then so does $g$).

The quarter-turn values and period give $\sin (\pi \mathrm{/}2+2m\pi )=1$ and $\sin (3\pi \mathrm{/}2+2m\pi )=-1$ for every integer $m$ (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi).

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

If two punctured-domain sequences approach a limit point while their images approach distinct real limits, then the function has no limit there (A function has no limit at $c$ as soon as two sequences in $A\setminus \{c\}$ tending to $c$ give different limits of the values).

The derivative at zero, if it exists, is the limit of $(f(x)-f(0))\mathrm{/}x$ as $x\to 0$ through nonzero reals (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set).

Sine is continuous; sums and products of continuous functions are continuous; quotients are continuous where their denominators do not vanish; and composites of continuous real functions are continuous (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, 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 composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs).

The number $\pi =2\gamma$ is positive because the first positive cosine zero 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).

Given: The function $F$ in the Example.

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

$(\sin u{)}^{\mathrm{\prime }}=\cos u$ (The derivatives of sine and cosine are cosine and minus sine).

The chain rule computes the derivative of a composite (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 $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Products and scalar multiples of differentiable functions are differentiable with the usual derivative formulas (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 $g(c)\mathrm{\ne }0$).

The power rule gives the derivatives of $x^2$ and $x^{-2}$ on their natural domains (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).

Sine vanishes and cosine equals $1$ at every integer multiple of $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi, The derivatives of sine and cosine are cosine and minus sine).

Every positive real has a unique positive square root (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$, case $n=2$).

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

Darboux integrability on $[a,b]$ is defined for bounded real functions on that interval (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$).

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

The derivative at zero is the limit of $(F(h)-F(0))\mathrm{/}h$ as $h\to 0$ through nonzero reals (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set).

Given: The universal claim in the Statement.

The function $F(0)=0$ and $F(x)=x^2\sin (1\mathrm{/}{x}^2)$ for $x\mathrm{\ne }0$ is differentiable on $\mathbb{R}$, with $F^{\mathrm{\prime }}(0)=0$, and $F^{\mathrm{\prime }}$ is unbounded on every neighbourhood of zero ($x^2\sin (1\mathrm{/}{x}^2)$ has an unbounded, non-Riemann-integrable derivative).

Given: The functions $f_k$ in the Example, on the compact interval $[0,\pi ]$ with its usual metric.

Sine is differentiable and hence continuous, and $\mathrm{\mid }\sin x\mathrm{\mid }\le 1$ for every real $x$ (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, Parity and the Pythagorean identity for sine and cosine).

The quarter-turn values and shift formulas determine $\sin (n\pi \mathrm{/}2)$ and give $\sin (\pi \mathrm{/}2)=1$ (Quarter-turn values and shifts by pi/2 and pi).

A family $\mathcal{F}$ is equicontinuous at $a$ when, for every $\epsilon >0$, one $\delta >0$ makes $d(x,a)<\delta$ imply $\mathrm{\mid }f(x)-f(a)\mathrm{\mid }<\epsilon$ for every $f\in \mathcal{F}$ (Equicontinuity, pointwise boundedness, and uniform boundedness for families in $C(K,\mathbb{R})$).

A uniform limit of continuous real functions is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).

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

For every continuous $g:[0,\pi ]\to \mathbb{R}$, ${\lim }_{n\to \mathrm{\infty }}{\int }_0^{\pi }g(x)\sin (nx)dx=0$ (Riemann–Lebesgue lemma for continuous functions on a compact interval).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$. Thus $[0,\pi ]$ is compact by Heine--Borel; real and metric compactness agree for its absolute-value subspace metric (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two, Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact, The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded, Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 5).

Affine real functions are continuous, and composites of continuous real functions are continuous (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, claim 5, A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs).

Given: The universal claim in the Statement.

For $f_k(x)=\sin ((k+1)x)$ on $[0,\pi ]$, the sequence $(f_k)$ is uniformly bounded, is not equicontinuous, and has no uniformly convergent subsequence (The sine harmonics are pointwise bounded but have no uniformly convergent subsequence).

Sine is continuous, affine real functions are continuous, and composites of continuous real functions are continuous (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, 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, claim 5, A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$, and every closed bounded interval in $\mathbb{R}$ is compact (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two, Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact).

Given: The graph $C:=\{(x,\sin (1\mathrm{/}x)):0<x\le 1\}$ and the set $S:=C\cup (\{0\}\times [-1,1])$ in ${\mathbb{R}}^2$.

Every interval in the real line, including $(0,1]$, is connected (The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$").

A continuous image of a connected subset is connected (A continuous image of a connected space is connected, and connectedness is a topological property).

A map into a product is continuous exactly when each component is continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, claim 2).

Sine has range $[-1,1]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

Sine has period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

The positive naturals are cofinal, and for every real $\epsilon >0$ some positive integer $N$ satisfies $1\mathrm{/}N<\epsilon$ (Every complete ordered field is Archimedean, For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

The reciprocal is continuous away from zero, sine is continuous, and composites of continuous real functions are continuous (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, The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs).

If $A$ is connected and $A\subseteq B\subseteq \bar{A}$, then $B$ is connected (If $A$ is connected and $A\subseteq B\subseteq \bar{A}$ then $B$ is connected; in particular the closure of a connected set is connected).

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).

Given: The set $S$ in the Statement and the two points $(0,0),(1,\sin 1)\in S$.

The topologist's sine curve $S=\{(x,\sin (1\mathrm{/}x)):0<x\le 1\}\cup (\{0\}\times [-1,1])$ is connected (The topologist's sine curve is connected).

Projections from a product are continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, claim 1).

For a continuous real function on a subset of $\mathbb{R}$, the preimage of a closed set is relatively closed ($f:A\to \mathbb{R}$ is continuous on $A$ if and only if the preimage of every open subset of $\mathbb{R}$ is the intersection with $A$ of an open subset of $\mathbb{R}$, and dually for closed sets); real and metric continuity agree for the usual metric (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 1).

The interval $[0,1]$ is compact (Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact), real and metric compactness agree on subsets of $\mathbb{R}$ (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 5), and a closed subset of a compact metric space is compact (A closed subset of a compact metric space is compact).

A continuous real function on a nonempty compact metric space attains its maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

A continuous real function on a connected space attains every intermediate value between two of its values (A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values).

The quarter-turn values and period give $\sin (\pi \mathrm{/}2+2m\pi )=1$ and $\sin (3\pi \mathrm{/}2+2m\pi )=-1$ for every integer $m$, and for every real $\epsilon >0$ some positive integer $N$ satisfies $1\mathrm{/}N<\epsilon$ (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi, For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

A space is path connected when every pair of points is joined by a continuous path from $[0,1]$ (Paths, path-connected spaces and path components).

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).

Every interval in the real line, including $[s,r]$, is connected (The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$").

A composite of continuous maps is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 1).

Given: The graph path $\gamma (x)=(x,\sin x)$ on $[0,2\pi ]$.

If $f:[a,b]\to \mathbb{R}$ has a continuous derivative $q$, then its graph has length ${\int }_a^b\sqrt{1+q(t{)}^2}dt$ (If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then the graph of $f$ has length ${\int }_a^b\sqrt{1+f^{\mathrm{\prime }}(t{)}^2}dt$).

$(\sin x{)}^{\mathrm{\prime }}=\cos x$ on $\mathbb{R}$ (The derivatives of sine and cosine are cosine and minus sine).

The supplementary, reflection, and quarter-turn identities give $\cos (\pi -x)=-\cos x$, $\cos (\pi \mathrm{/}2-x)=\sin x$, and $\cos (-x)=\cos x$ (Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions).

Integrals are additive over subintervals, and affine substitutions obey the substitution formula with oriented endpoints (For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$, Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$, The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$).

For every real $x$, ${\sin }^{2}x+{\cos }^{2}x=1$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains).

Every positive real has a unique positive square root (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$, case $n=2$).

Differentiable real functions are continuous (A function differentiable at $c$ is continuous at $c$).

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).

For $\alpha \in \mathbb{R}$, the function $u\mapsto u^{\alpha }$ is continuous on $(0,\mathrm{\infty })$; in particular the positive square-root function is continuous there (Continuity and derivatives of positive-base real powers).

Polynomial algebra and composites preserve continuity, and continuous functions on a compact interval are Riemann integrable (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 composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

Given: The interval $I:=(-\pi \mathrm{/}2,\pi \mathrm{/}2)$ and the usual absolute-value metrics on $I$ and $\mathbb{R}$.

Tangent restricts to a continuous strictly increasing bijection $\tan :I\to \mathbb{R}$, whose inverse $\arctan :\mathbb{R}\to I$ is continuous (The principal inverse tangent $\arctan :\mathbb{R}\to (-\pi \mathrm{/}2,\pi \mathrm{/}2)$).

A homeomorphism is a continuous bijection with continuous inverse (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

In the usual metric, $B(0,r)=(-r,r)$ for $r>0$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Every Cauchy sequence of real numbers converges to a real number (The reals are complete).

For a subset of $\mathbb{R}$ with the subspace metric, metric Cauchy sequences are exactly real Cauchy sequences (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 7).

A subset of a metric space is closed exactly when it is sequentially closed (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed, claim 2).

A subspace of a complete metric space is complete if and only if it is closed (A subspace of a complete metric space is complete iff it is closed, and a complete subspace of any metric space is closed).

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

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).

The usual real line is not a bounded metric space (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

A sequence converges in a metric space when its distance from the proposed limit is eventually below every positive tolerance (Convergence of a sequence in a metric space: $x_k\to x$ iff $d(x_k,x)\to 0$ in $\mathbb{R}$); for the usual real metric this distance is $\mathrm{\mid }{x}_k-x\mathrm{\mid }$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

A metric space is complete when every Cauchy sequence in it converges to a point of the space (Complete metric space: every Cauchy sequence converges in the space).

A subset of a metric space is bounded when it is empty or is contained in some open ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Given: The zero-based function series ${\sum }_{k=0}^{\mathrm{\infty }}{f}_k(x)$ with $f_k(x):=\sin ((k+1)x)\mathrm{/}(k+1)$.

If $x\mathrm{\notin }2\pi \mathbb{Z}$, then for every positive integer $N$, $\mathrm{\mid }{\sum }_{n=1}^N\sin (nx)\mathrm{\mid }\le 1\mathrm{/}\mathrm{\mid }\sin (x\mathrm{/}2)\mathrm{\mid }$ (Finite sums of the sine harmonics).

If the partial sums of $(a_k)$ are bounded and $(b_k)$ is nonincreasing with limit zero, then $\sum a_k{b}_k$ converges (Dirichlet's test: if the partial sums of $\sum a_k$ are bounded and $(b_k)$ is nonincreasing with $b_k\to 0$, then $\sum a_k{b}_k$ converges).

Sine vanishes at every integer multiple of $\pi$ and has period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

Uniform convergence of a function series is equivalent to the uniform Cauchy condition on every sufficiently late finite tail (A series of real-valued functions converges uniformly if and only if its tails are uniformly small).

For $0<t\le 2$, $\sin t\ge t\mathrm{/}3>0$ (Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3).

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