# Real numbers/Completeness/Nested intervals/Roots/Section

## Definition

A sequence of closed intervals

in is called
(a sequence of)
*nested intervals*, if
holds for all
,
and if the sequence of the lengths of the intervals, i.e.

In a family of nested intervals, the length of the intervals are a decreasing null sequence. However, we do not require a certain velocity of the convergence. An *interval bisection* is a special kind of nested intervals, where the next interval is either the lower or the upper half of the preceding interval.

## Theorem

Suppose that , , is a sequence of nested intervals in . Then the intersection

contains exactly one point . Nested intervals determine a unique real number.

### Proof

## Theorem

For every nonnegative real number and every there exists a unique nonnegative real number fulfilling

### Proof

We define recursively nested intervals . We set

and we take for an arbitrary real number with . Suppose that the interval bounds are defined up to index , the intervals fulfil the containment condition and that

holds. We set

and

Hence one bound remains and one bound is replaced by the arithmetic mean of the bounds of the previous interval. In particular, the stated properties hold for all intervals and we have a sequence of nested intervals. Let denote the real number defined by this nested intervals according to fact. Because of exercise, we have

Due to fact, we get

Because of the construction of the interval bounds and due to fact, this is but also , hence .

This uniquely determined number is denoted by or by .