There is another post with this exact same prompt which got several down-votes for not showing their work. So I'll show what work I've got. I know being a harmonic function implies satisfying the Mean Value Property, thus what I thought I'd do is consider two arbitrary points in $\mathbb{C}$ and prove that:
$|u(z) - u(w)|=|\frac{1}{2\pi}\int_0^{2\pi}u(z - re^{i\theta})d\theta - \frac{1}{2\pi}\int_0^{2\pi}u(w - te^{i\theta})d\theta| $ $= |\frac{1}{2\pi}\int_0^{2\pi}u(z - re^{i\theta}) - u(w - te^{i\theta})d\theta|$
Now what I wan't to do is take r and t to infinity and prove this integral goes to zero, but I can't figure out how to do that. I've used the fact that my harmonic function u is defined on all of $\mathbb{C}$ in taking t and r to infinity but I still need to use the fact that it's bounded.
I also read through the proof of Liouville's Theorem, given as an answer in that other post, which seems intuitively correct but rather un-rigorous. It actually says that the value in the center of a ball is equal to the average over the ball's volume. I imagine this can be proven from the average over the boundary since the average over the boundary will not change as you continuously shrink the boundary down to its center. This however would require the difference of two surface integrals, or is there a simpler way?
I was also considering using the Maximum Modulus Principle somehow but I haven't found a way yet. I've probably made this way more complicated than it has to be, hopefully someone can help me, thanks! =].
Edit: Ok after thinking some more about the accepted answer in that other post, I've decided what I need to do is look at the canonical holomorphic function whose real part is my harmonic function u, and prove that holomorphic function is bounded. From there I can prove that since it's entire and bounded that it's constant, and from there prove that then its real part must be constant. So could someone talk a bit about the construction of this canonical holomorphic function?