In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals ${\displaystyle I_{n}}$ on the real number line with natural numbers ${\displaystyle n=1,2,3,\dots }$ as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:

1. Every interval in the sequence is contained in the previous one (${\displaystyle I_{n+1}}$ is always a subset of ${\displaystyle I_{n}}$).
2. The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold ${\displaystyle \varepsilon }$ after a certain index ${\displaystyle N}$).

In other words, the left bound of the interval ${\displaystyle I_{n}}$ can only increase (${\displaystyle a_{n+1}\geq a_{n}}$), and the right bound can only decrease (${\displaystyle b_{n+1}\leq b_{n}}$).

Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and surcumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi (${\displaystyle \pi }$).

The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval ${\displaystyle I_{n}}$ (thus, for all ${\displaystyle n\in \mathbb {N} }$). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers).