(a) Suppose $u$ is harmonic in a punctured ball $B\left(z_0, r\right) \backslash\left\{z_0\right\}$. Prove
$$
\lim _{z \rightarrow z_0} \frac{u(z)}{\log \left|z-z_0\right|}=0
$$
if and only if $u$ extends to be harmonic in $B\left(z_0, r\right)$. This is the "harmonic" version of Riemann's removable singularity theorem. In particular, if $u$ is bounded and harmonic on the punctured ball, then $u$ extends to be harmonic on $B(\hbar 0, r)$.
(b) Suppose $u$ is non-constant and harmonic in $\mathbb{C}$. Prove
$$
\lim _{r \rightarrow \infty} \frac{M(r)}{\log r}=\infty,
$$
where $M(r)=\sup _{|z|=r} u(z)$.