Constant bounded entire function by comparing two values

Suppose $f$ is analytic at every point of $\mathbb{C}$ and $|f|\le M$ everywhere. Without quoting Liouville's theorem, prove that $f(a)=f(b)$ for any two points $a,b$.

Use Cauchy's integral formula on a large circle. For $|z|=R$ large, $f(a)-f(b)=\frac{a-b}{2\pi i}\oint\frac{f(z)}{(z-a)(z-b)}\,dz$. The integrand is $O(M/R^2)$ on a contour of length $2\pi R$, so the right side is $O(M|a-b|/R)\to0$, forcing $f(a)=f(b)$.