Proving Liouville's Theorem Using Cauchy Integral Formula

[eaglesmath15]

Homework Statement

Prove Liouville's theorem directly using the Cauchy Integral formula by showing that f(z)-f(0)=0.

Homework Equations

f(a) = $\frac{1}{2πi}$$\oint\frac{f(z)}{z-a}dz$

The Attempt at a Solution

So the thing is, I know how to prove Liouville's theorem using CIF, but it doesn't show f(z)-f(0)=0, or at least not directly, and I've tried looking up other methods of proving it this way, but can't find any.

 

[brmath]

The proof I know expands f into a Taylor's series at zero , and shows that each $a_k$ has to be zero except for k = 0. We know $a_0$ = f(0). Are you familiar with this approach?