Liouville's Theorem is as follows:
Liouville's Theorem. If $f$ is holomorphic and bounded on $\mathbb C$, then it is constant.
Proof. We have the following formula for the derivative of the function $f$: where $\mathrm{ind}(\gamma,z)=1$. If we suppose that $f$ is bounded in magnidue so that $|f(z)|
0$, and we choose $\gamma$ to be a circle of radius $R$ centered at $z$, the integral identity can be transformed into the following bound: but because $R$ can be chosen to be arbitrarily large, we have that $|f'(z)| 0$, and hence $f'(z)$ vanishes for all $z$, implying that $f$ is constant. $\blacksquare$