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.