= Closed real interval
= Closed interval
{synonym}
A bounded <closed interval> includes both of its endpoints:
$$
[a,b]=\{x\in\mathbb R:a\leq x\leq b\},\qquad a\leq b.
$$
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>.
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