Property of the Green's Function I am trying to prove a statement about Green's Functions but can't seem to conclude anything. The statement is this:

Let $G_1$ be the Green's Function of the Laplacian $\Delta$ on $B(0,1)\subset \mathbb R^3$. Let $G_2$ be the Green's Function of the Laplacian $\Delta$ on $B(0,2)\subset \mathbb R^3$. $B(0,1)$ is the Ball centered at the Origin with radius $1$, same analogous statement for $B(0,2)$. Let $x_0\in B(0,1)$. Prove that for all $x\in B(0,1)$ such that $x\neq x_0$, we have:$$G_1(x,x_0)>G_2(x,x_0)$$

So the way I tried to prove this is by considering a region $B(0,1)_\epsilon=B(0,1)\setminus B(0,\epsilon)$ for some $1>\epsilon>0$. Then, $G_1$ is certainly Harmonic on $B(0,1)_\epsilon$. Thus, we can apply the Maximum Property to show that $G_1$ reaches its maximum $M$ on $\partial B(0,1)$. By the minimum Principle, it is also bounded below by $m$. Thus, we see that $m<G_1(x,x_0)<M$.
Now, applying the same region for $B(0,2)$, we get $B(0,2)_\epsilon$. Applying the Minimum Principle and Maximum Principle again (noting that $\partial B(0,1)$ lies within $B(0,2)$, we find that $G_2$ reaches its maximum $N$ and minimum $n$ on $\partial D(0,2)$. Therefore, everywhere on $B(0,2)_\epsilon$, we have that $n<G_2<N$. 
I proved beforehand that $G(x,x_0)$ is negative on some domain $\Omega$. I feel like the conclusion to this proof is right in front of me but I can't seem to string these two facts together. If someone has the next step for this proof, I would appreciate it greatly. Thank you!

By definition, for each $x_0 \in B_2$ Green's function $G_2$ is the solution of 
$$
\begin{cases}
\Delta_x G_2 (x,x_0) = \delta_{x_0} & \text{for } x \in B_2, \\
G_2 (x,x_0) = 0 & \text{for } x \in \partial B_2.
\end{cases}
$$
In the above, $G_2(\cdot,x_0)$ is considered as a function of the first variable and the first equation is meant in distributional sense. The same holds for $G_1$ in $B_1$. 
As you already observed, $G_2(x,x_0) < 0$ for $x \in B_2$, $x \neq x_0$. Choose $x_0 \in B_1$ and consider the function $g(x) = G_2(x,x_0) - G_1(x,x_0)$. From what we know about $G_1$ and $G_2$, we conclude that it satisfies 
$$
\begin{cases}
\Delta g(x) = 0 & \text{for } x \in B_1, \\
g(x) < 0 & \text{for } x \in \partial B_1.
\end{cases}
$$
By maximum principle, $g(x) < 0$ for all $x \in B_1$. 

Note that the above reasoning applies not only to balls, but arbitrary domains $\Omega_1 \subsetneq \Omega_2$.
