So I was wondering if I could prove something about Harmonic function in an alternative and rather simple seeming way. Liouville's theorem says that if a function is harmonic on all of $\mathbb{R}^d$, and is bounded, then it is constant. The proof I have always seen is one that establishes bounds on the norm of the gradient of the function in question (in terms of the radius of some ball on which the function is defined), and then lets the radius of the ball tend to infinity. I was thinking something different.
Take the point where $u$ achieves its maximum. Then, take a ball of arbitrary radius $R$ around it. We know $u$ is harmonic on this domain, hence by the strong maximum principle, because $u$ achieves its maximum on this ball, $u$ is constant on this ball (and equal to its maximum). Let the radius of the ball go to infinity. This proves the result. What is wrong with this method of proof?