I'm trying to prove 2.74 Theorem of Folland's book, Introduction to PDE. It says:
If $u$ is harmonic on the complement of a bounded set in $\mathbb{R}^n$, the following are equivalent:
a) $u$ is harmonic at infinity.
b) $u(x)\to0$ as $x\to\infty$ if n>2, or $|u(x)|=o(log|x|)$ as $x\to\infty$ if $n=2$.
c)$|u(x)|=O(|x|^{2-n })$ as $x\to\infty$.
I've proved a) implies b) and c) implies a) but I'm stuck proving b) implies c).
I started with n=2, it's required to prove $|u(x)|=O(1)$. The assumption means that for every $\epsilon >0$ exists $M>0$ such that $ |u(x)|<\epsilon\, log|x|$ for every $|x|\geq M.$ From this, ¿How can I prove that $|u(x)|<c$ for every $|x|\geq |x_0|$ for some $C>0$ and some $x_0$?.
For $n>2$, I have the same problem. I think I'm missing some theorem for harmonic functions. I was thinking about maximum principle or maybe the fact that Kelvin transform is harmonic in a punctured bounded set, but I'm not sure how to use them. Please, give me some hint for this implication or tell me if I'm using wrong something.