Let
be
real numbers,
and let
denote a
continuous function
such that
and .
Then the function has a zero within the interval, due to
the Intermediate value theorem.
Such a zero can be found by the bisection method, as described in the proof
of the Intermediate value theorem.
We put
and ,
and the other interval bounds are inductively defined in such a way that
and
hold. Define
and compute . If
,
then we set
-
and if
,
then we set
-
In both cases, the new interval has half the length of the preceding interval and so we have bisected intervals. The
real number
defined by these nested intervals is a zero of the function.