The Nested Interval Theorem in Higher Dimensions

Nested Intervals

Definition: A sequence of intervals $I_{n}$ where $n\in \mathbb {N}$ is said to be nestedif $I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq ...\supseteq I_{n}\supseteq I_{n+1}\supseteq ...$.

For example, consider the interval $I_{n}=[0,{\frac {1}{n}}]$ We note that $I_{1}=[0,1]$, $I_{2}=[0,{\frac {1}{2}}]$, $I_{3}=[0,{\frac {1}{3}}]$, … As we can see $I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq ...\supseteq I_{n}\supseteq I_{n+1}\supseteq ...$ and so the sequence of intervals $I_{n}$ is nested. The diagram above illustrates this specific nesting of intervals.

Sometimes a nested interval will have a common point. In this specific example, the common point is $0$ since $0\leq 0<{\frac {1}{n}}$ for all $n\in \mathbb {N}$. We denote the set theoretic intersection of all these intervals to be the set of common points in a nested set of intervals:

${\begin{aligned}\bigcap _{n=1}^{\infty }I_{n}=C\end{aligned}}$

Sometimes a set of nested intervals does not have a common point though. For example consider the set of intervals $I_{n}=(n,\infty )$. Clearly $I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq ...\supseteq I_{n}\supseteq I_{n+1}\supseteq ...$ since $I_{1}=(1,\infty )$, $I_{2}=(2,\infty )$, … However, there is no common point for these intervals.

Example 1

Determine the set $C$ of points to which the set of nested intervals $I_{n}=(1-{\frac {1}{n}},2+{\frac {1}{n}})$ have in common.

We first note that $I_{1}=[0,3]$, $I_{2}=[0.5,2.5]$, $I_{3}=[0.66...,2.33...]$, … We conjecture that the set of points $C=\bigcap _{n=1}^{\infty }I_{n}=[1,2]$ are contained within all the intervals. This can be informally deduced since $\lim _{n\to \infty }1-{\frac {1}{n}}=1$ and $\lim _{n\to \infty }2+{\frac {1}{n}}=2$.

The Nested Intervals Theorem

We have just looked at what exactly a Nested Interval is, and we are about to look at a critically important theorem in Real Analysis. Before we look at the Nested Intervals Theorem let's first look at the following important lemma that will be used to prove the Nested Intervals Theorem.

Licensing

Content obtained and/or adapted from:

• Nested Intervals, mathonline.wikidot.com under a CC BY-SA license