---
abstract: 'We present a short proof of Cantor’s Theorem (circa 1870s): if $a_n \cos nx + b_n \sin nx \to 0$ for each $x$ in some (nonempty) open interval, where $a_n, b_n$ are sequences of complex numbers, then $a_n$ and $b_n$ converge to 0.'
address: 'Department of Mathematics & Statistics, University of Northern B.C., Prince George, B.C. V2N 4Z9, Canada.'
author:
- 'S. W'
date: 'March 30, 2020'
title: A P O C T
---

Proof Of Cantor’s Theorem
=========================

[**Cantor’s Theorem.**]{} Let $a_n, \, b_n$ be sequences of complex numbers such that $$\lim_{n\to\infty} \ a_n \cos nx + b_n \sin nx \ = \ 0$$ for each $x$ in some open interval $(c,d)$. Then $a_n \to 0$ and $b_n \to 0$.

The proof presented here consists of reduction to the case $C_n  \sin nx \to 0,$ which is covered by Lemma B below and which we proceed to prove first.

[**Lemma A.**]{} Let $\delta > 0 $ be given and let $B_n$ be a [*bounded*]{} sequence of complex numbers such that $B_n \sin n t \to 0$ as $n\to \infty$ for each $0 < t < \delta$. Then $B_n \to 0$.

Replacing $t\in(0,\delta)$ by $2t$ we have $$B_n \sin (n 2t) = 2 B_n \sin(nt) \cos(nt) \to 0$$ so the limit $B_n \sin n x \to 0$ holds also for $0 < x < 2\delta$. By repeating this doubling procedure a finite number of times, we get $B_n \sin n x \to 0$ for $0 \le x \le 2\pi \le N\delta$ for some fixed positive integer $N$. Therefore, $f_n(x) := B_n \sin n x \to 0$ for each $x$ in the closed interval $[0,2\pi]$. Further, since the sequence $B_n$ is bounded, the functions $f_n$ are bounded by an (integrable) constant on $[0,2\pi]$. Therefore, by the Dominated Convergence Theorem (, Theorem 11.32) $$0 \ = \ \lim_n \rmintop_0^{2\pi} |f_n(x)| dx \ =\  \lim_n \ |B_n|  \rmintop_0^{2\pi} |\sin nx| dx 
\ = \  \lim_n \ 4 |B_n|$$ since $\int_0^{2\pi} |\sin nx| dx = 4$ for each integer $n\ge1$. It follows that $B_n \to 0$.

The next lemma removes the boundedness condition in Lemma A.

[**Lemma B.**]{} Let $\delta > 0 $ be given and let $C_n$ be a sequence of complex numbers such that $C_n \sin n t \to 0$ as $n\to \infty$ for each $0 < t < \delta$. Then $C_n \to 0$.

Consider the bounded sequence $B_n = \frac{ |C_n| }{ 1+|C_n|},$ for which we have $$\lim_{n} \ B_n \sin nx \ =\ \lim_{n} \ \frac{ |C_n| \sin nx}{ 1+|C_n|}  = 0$$ holds for each $x$ in $(0,\delta)$. Hence $B_n \to 0$ by Lemma A, which in particular means that there is an integer $M$ such that $B_n < \tfrac12$ for $n > M$. This says that $|C_n| < 1$ (for $n>M$) so that $C_n$ is bounded. By Lemma A, applied again to $C_n,$ we see that $C_n\to 0$.

[*Proof of Cantor’s Theorem.*]{} We are given that $$a_n \cos nx + b_n \sin nx \to 0$$ for each $c < x < d$. Let $\delta > 0$ such that $x + \delta < d,$ so that $c < x+t < d$ holds for each $0 \le t < \delta$. Then since the zero limit holds also for $x + t,$ we have $$\begin{aligned}
\label{lastline}
a_n \cos n&(x+t) + b_n \sin n(x+t) = a_n \cos(nx+nt) + b_n \sin(nx+nt)  \notag
\\
&= a_n \big[ \cos(nx) \cos(nt) - \sin(nx)\sin(nt)  \big] + b_n \big[ \sin(nx) \cos(nt) + \cos(nx) \sin(nt)  \big]   \notag
\\
&= \big[ a_n \cos(nx) + b_n \sin(nx) \big] \cos(nt) 
+ \big[ b_n \cos(nx) - a_n \sin(nx) \big] \sin(nt).\end{aligned}$$ Since this whole sum goes to 0, and since the first term in goes to 0 by hypothesis, it follows that the second term goes to 0, i.e. $C_n \sin(nt) \to 0$ for each $0 < t < \delta$ where $C_n := b_n \cos(nx) - a_n \sin(nx)$ is a fixed sequence (independent of $t$) since here $x$ is fixed. Lemma B above now applies to this and gives $$C_n = b_n \cos(nx) - a_n \sin(nx) \ \to \ 0$$ as $n\to\infty$. Therefore taking the sum of absolute squares we get $$\big|a_n \cos(nx) + b_n \sin(nx)\big|^2 + 
\big|b_n \cos(nx) - a_n \sin(nx)\big|^2 = |a_n|^2 + |b_n|^2$$ which has zero limit, hence the result.

[10]{}

G. Cantor, [*Uber einen die trigonometrischen Reihen betreffenden Lehrsatz*]{}, Crelles Journal für Mathematik, Bd. [**72**]{} (1870), 130-138. R. L. Cooke, *The Cantor-Lebesgue Theorem*, American Mathematical Monthly [**86**]{}, No. 7, 558-565.

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, Third Ed. 1976.
