Nested intervals

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a sequence of nested intervals can be intuitively understood as an ordered collection of <a href="/facts/Interval_(mathematics)/e9se3vTe">intervals</a> 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
 on the <a href="/facts/Interval_(mathematics)/e9se3vTe">real number line</a> with <a href="/facts/Natural_number/u65og8JE">natural numbers</a> 
 
 
 
 n
 =
 1
 ,
 2
 ,
 3
 ,
 …
 
 
 {\displaystyle n=1,2,3,\dots }
 
 as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:

<ol><li>Every interval in the sequence is contained in the previous one (
 
 
 
 
 I
 
 n
 +
 1
 
 
 
 
 {\displaystyle I_{n+1}}
 
 is always a subset of 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
).</li>
<li>The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 after a certain index 
 
 
 
 N
 
 
 {\displaystyle N}
 
).</li></ol>
In other words, the left bound of the interval 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
 can only increase (
 
 
 
 
 a
 
 n
 +
 1
 
 
 ≥
 
 a
 
 n
 
 
 
 
 {\displaystyle a_{n+1}\geq a_{n}}
 
), and the right bound can only decrease (
 
 
 
 
 b
 
 n
 +
 1
 
 
 ≤
 
 b
 
 n
 
 
 
 
 {\displaystyle b_{n+1}\leq b_{n}}
 
).
Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient <a href="/facts/Babylonia/J4jjdUua">Babylonians</a> discovered a <a href="/facts/Methods_of_computing_square_roots/dRG2BtBp">method for computing square roots</a> of numbers. In contrast, the famed <a href="/facts/Archimedes/sIeMhSVF">Archimedes</a> constructed sequences of polygons, that inscribed and circumscribed a unit <a href="/facts/Circle/9fy5ggKn">circle</a>, in order to get a lower and upper bound for the circles circumference - which is the <a href="/facts/Pi/vC0UBj1q">circle number Pi</a> (
 
 
 
 π
 
 
 {\displaystyle \pi }
 
).
The central question to be posed is the nature of the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
 (thus, for all 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of rational numbers).

Nested intervals open-in-new

Nested intervals