The proof of Theorem 4.1 is very straight-forward. An immediate proof involves induction and an argument which shows that the sequence of approximations is a Cauchy sequence. The usefulness of Theorem 4.1 is two-fold: First, Theorem 4.1 states that convergence is guaranteed; an attribute not commonly found in algorithms. Secondly, it provides a definitive convergence criterion which may be used to determine the maximum number of iterations prior to any calculations.
To see how many iterations will be required for convergence, Equation 4.1 can be used. The bisect method's convergence condition is:
Equation 4.1 says that this will be satisfied when
or,
The quantity of interest here is n, which can be solved for after taking logarithms of both sides, yielding
In addition, Equation 4.1 also provides us the necessary result for the the bisection method's order of convergence. Recalling Definition 1.7, we have
