Cantor's Nested Intervals Theorem 📂Analysis

Cantor's Nested Intervals Theorem

Definition¹

A sequence $\left\{ S_{n} \right\}_{n=1}^{\infty}$ of sets is said to be nested if $S_{n+1} \subset S_{n}$ for every natural number $n$.

Explanation

The Korean translation of “nested” is not very smooth, and since there is no particularly good alternative, it is recommended to simply memorize it as “Nested”.

Theorem

For nested intervals $[a_{n}, b_{n}]$, the following hold.

(a) $\displaystyle \bigcap_{n=1}^{\infty} [a_{n}, b_{n}] \ne \emptyset$

(b) In particular, if $\displaystyle \lim_{n \to \infty} (b_{n} - a_{n}) = 0$, then $\displaystyle \bigcap_{n=1}^{\infty} [a_{n}, b_{n}]$ is a singleton set.

A singleton set refers to a set that has only one element.

Proof

(a)

By assumption, for every natural number $n$,

$$ [a_{n+1} , b_{n+1} ] \subset [a_{n} , b_{n} ] \\ a_{1} \le a_{n} \le b_{n} \le b_{1} $$

By the completeness axiom, the two numbers

$$ a:=\sup \left\{ a_{n} \right\} \\ b:=\inf \left\{ b_{n} \right\} $$

exist. Since $a_{n} \le a \le b \le b_{n}$ holds for every natural number, we have $[a,b] \subset [a_{n} , b_{n} ]$, and therefore

$$ \bigcap_{n=1}^{\infty} [a_{n}, b_{n}] \ne \emptyset $$

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(b)

Assuming $\displaystyle \lim_{n \to \infty} (b_{n} - a_{n}) = 0$, we have $a=b$, so

$$ \bigcap_{n=1}^{\infty} [a_{n}, b_{n}] = \left\{ a \right\} = \left\{ b \right\} $$

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