known (starting at ), one can define

To put this into words, one can compare the midpoint of to in order to determine whether the midpoint is smaller or larger than . If the midpoint is smaller, one can set it as the lower bound of the next interval , and if the midpoint is larger, one can set it as the upper bound of the next interval. This guarantees that . With this construction the intervals are nested and their length get halved in every step of the recursion. Therefore, it is possible to get lower and upper bounds