= Nested intervals
{wiki}
A sequence of <nested intervals> keeps every new enclosure inside the previous one:
$$
I_1\supseteq I_2\supseteq I_3\supseteq\cdots.
$$
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.
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