Cantor's Intersection Theorem 📂Analysis

Cantor's Intersection Theorem

Definition¹

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

Explanation

The translation of nested might not be smooth, but since there is no better alternative, it is recommended to just memorize it as “Nested.”

Theorem

For the nested interval $[a_{n}, b_{n}]$, the following holds:

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

(b) Specifically, 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 is a set that contains only one element.

Proof

(a)

Given that for all natural numbers $n$

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

by the axiom of completeness, there are two numbers

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

Since for all natural numbers, $a_{n} \le a \le b \le b_{n}$ holds, it follows that $[a,b] \subset [a_{n} , b_{n} ]$, hence

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

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(b)

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

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

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