Suppose that
|*f*(*z*)| < *M* and that Cauchy's integral formula is used to compare
the values of *f*(*z*) at two points. Then

f(z_{2}) - f(z_{1}) |
= | (151) |

A little rearrangement produces

= | (152) |

which could very easily be used in an alternative derivation of the Cauchy formula for a derivative. Now, choose a huge circular contour centered on the auxiliary point

But in the difference we are calculating, the common denominator introduces an additional power of *r*, leaving little doubt that the combination
approaches zero. Very well, take the limit, and arrive at the conclusion a bounded analytic function can only be a constant. That is the content of *Liouville's theorem*.