Khanh at 22:29, 13 November 2021

2021-11-13T22:29:23Z

Khanh: Created page with "== Nested Intervals ==
:'''Definition:''' A sequence of intervals $I_n$ where $..."$

2021-11-13T21:43:28Z

Created page with "== Nested Intervals == <blockquote style="background: white; border: 1px solid black; padding: 1em;"> :'''Definition:''' A sequence of intervals <math>I_n</math> where <math>..."

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== Nested Intervals ==
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Definition:''' A sequence of intervals <math>I_n</math> where <math>n \in \mathbb{N}</math> is said to be '''[http://mathonline.wdfiles.com/local--files/nested-intervals/Screen%20Shot%202014-10-05%20at%207.37.43%20PM.png nested]'''if <math>I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...</math>.
</blockquote>

For example, consider the interval <math>I_n = [0, \frac{1}{n} ]</math> We note that <math>I_1 = [0, 1]</math>, <math>I_2 = [0, \frac{1}{2}]</math>, <math>I_3 = [0, \frac{1}{3} ]</math>, … As we can see <math>I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...</math> and so the sequence of intervals <math>I_n</math> 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 <math>0</math> since <math>0 \leq 0 < \frac{1}{n}</math> for all <math>n \in \mathbb{N}</math>. We denote the set theoretic intersection of all these intervals to be the set of common points in a nested set of intervals:

<div style="text-align: center;"><math>\begin{align} \bigcap_{n=1}^{\infty} I_n = C \end{align}</math></div>

Sometimes a set of nested intervals does not have a common point though. For example consider the set of intervals <math>I_n = (n, \infty)</math>. Clearly <math>I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...</math> since <math>I_1 = (1, \infty)</math>, <math>I_2 = (2, \infty)</math>, … However, there is no common point for these intervals.

===Example 1===
'''Determine the set <math>C</math> of points to which the set of nested intervals <math>I_n = (1 - \frac{1}{n}, 2 + \frac{1}{n} )</math> have in common.'''

We first note that <math>I_1 = [0, 3]</math>, <math>I_2 = [0.5, 2.5]</math>, <math>I_3 = [0.66..., 2.33...]</math>, … We conjecture that the set of points <math>C = \bigcap_{n=1}^{\infty} I_n = [1, 2]</math> are contained within all the intervals. This can be informally deduced since <math>\lim_{n \to \infty} 1 - \frac{1}{n} = 1</math> and <math>\lim_{n \to \infty} 2 + \frac{1}{n} = 2</math>.

== 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:
* [http://mathonline.wikidot.com/nested-intervals Nested Intervals, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 20: / Line 20: @@
 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:
 * [http://mathonline.wikidot.com/nested-intervals Nested Intervals, mathonline.wikidot.com] under a CC BY-SA license

The Nested Interval Theorem in Higher Dimensions - Revision history

Khanh at 22:29, 13 November 2021

Khanh: Created page with "== Nested Intervals == :'''Definition:''' A sequence of intervals I_n where ..."

Khanh: Created page with "== Nested Intervals ==
:'''Definition:''' A sequence of intervals $I_n$ where $..."$