Facts that would be helpful to the proof:
I honestly have no idea how to begin/proceed since we rarely look at proofs or abstract idea "near the origin". We have only done a lot of parameterizations and flux integrals, etc.
Any help or sketch of proof would be appreciated!! :)
I will just give you an idea for showing that $||\nabla g||^2 = 0 $ inside a sufficiently small sphere $S_1$, which means $g$ is a constant inside the sphere (near the origin).
Part (a) is nothing but the statement that $f$ is a minimum at the origin,
Part (b) is the Green's first identity, roughly proven by using integration by parts.
Now, $g$ is a harmonic function, $\Delta g= 0$, then $f=g-C$ is also a harmonic function, where $C$ is a positive constant which is large enough to make $f<0$ inside the region $S_1$
Using the identity, from part (b) and $\Delta g=\Delta f= 0$, $$\int_{S_1} f \nabla f \cdot d\vec{S} = \int_{x\in S_1}||\nabla g||^2 dV$$
The right hand side is obviously $\ge 0$, what about the left hand side? (remember that by construction $g<0$ for $x \in S_1$)
I think you can figure this out by yourself. :)