Interval (mathematics)

A real interval contains every real number lying between any two of its elements.

An interval that includes exactly one endpoint, such as $[a,b)=\{x:a\le x<b\}$. Adjacent half-open intervals $[a,b)$ and $[b,c)$ are disjoint while covering $[a,c)$, which is useful for representing adjoining occupied cells without treating shared boundaries as overlaps.

An interval contained in another interval.

A sequence of nested intervals keeps every new enclosure inside the previous one:Each step adds a restriction without discarding an earlier restriction. This is useful when an exact real number is described by increasingly precise bounds rather than by a finite decimal expansion. The bisection method, for instance, produces nested intervals that keep enclosing a zero of a function.

For a simple example, the closed intervals $I_n=[1-1/n,1+1/n]$ are nested and all contain $1$. Their lengths $2/n$ tend to zero, so the nested interval theorem leaves exactly that point. The family $J_n=[1-1/n,2+1/n]$ is also nested, but its intersection is the whole closed interval $[1,2]$. Nesting alone does not force uniqueness.

The open intervals $K_n=(0,1/n)$ illustrate a different possibility. Every finite collection has a nonempty intersection, but the intersection of all of them is empty: any positive candidate eventually exceeds $1/n$, and zero is excluded from every member. Each finite row in the diagram contains points; the full infinite family need not.

If $I_1\supseteq I_2\supseteq \cdots$ are nonempty closed intervals with finite endpoints, then at least one real number belongs to every member. If their lengths tend to zero, exactly one real number belongs to every member. The result turns an unlimited sequence of consistent enclosures into an existence guarantee.

Write $I_n=[a_n,b_n]$. Nesting makes the left endpoints nondecreasing and the right endpoints nonincreasing. Their bounds give finite supremum and infimum, withIndeed, every $b_n$ is an upper bound for all the $a_j$, so the least-upper-bound property gives ${\sup }_j{a}_j\le b_n$ for every $n$. This proves the displayed interval is nonempty. If $b_n-a_n\to 0$, its two endpoints must coincide. This is the connection to the completeness of the real numbers discussed in Günther Hörmann and David Langer, Einführung in die Analysis, §§3.15–3.16: nested intervals and order completeness.

The distinction between existence and uniqueness is useful in computation. If an unknown point $x_{\ast }$ lies in every $[a_n,b_n]$, the midpoint $m_n=(a_n+b_n)/2$ satisfiesThus the enclosure itself certifies an absolute error without knowing the answer. The bisection error bound is the special case where each update halves the width.

Closedness and a finite initial enclosure matter. The open intervals $(0,1/n)$ have no common point, while the closed but unbounded intervals $[n,\infty )$ also have empty intersection. A picture of a few intervals cannot certify the intersection of an infinite family; the hypotheses of the nested interval theorem do that work.

A bounded closed interval includes both of its endpoints:If $a=b$, it consists of one point. Its length is $b-a$. In contrast, the open interval $(a,b)$ excludes both endpoints; half-open real intervals include just one.

Including the endpoints has a useful consequence for limits: a convergent sequence whose terms lie in $[a,b]$ has its limit of a sequence in $[a,b]$ too. Finite endpoints also make this a bounded set, so it is a compact set by the Heine-Borel theorem. These two properties support both the nested interval theorem and the extreme value theorem.

A compact interval in the real numbers is either the empty set or a closed bounded interval $[a,b]$ with $a\le b$. Its compactness follows from the Heine-Borel theorem.