Proof for Mean Value Property using a specific limit I am trying to prove the following:

Suppose $u \in C^2(\Omega)$. For some $x \in \Omega$ we have that
\begin{align} \Delta u(x) = \lim_{r \to 0} \frac{2n}{r^2} \left[
 \frac{1}{w_n} \int_{\partial B_{1}} u(x + ry)\mathrm{d}S_y - u(x) 
 \right]. \end{align}
If $u$ is harmonic, then $u$ has the Mean Value Property.


I believe that we basically need to see this limit in some smart way, but I still couldnt do that. Can someone give me a hint on this? But it must use the limit above. 
Thanks in advance.
 A: Version 1
Maybe stepping back one step actually make things clearer. 
You want to proof that if $\Delta u(x) \equiv 0$, then for any $x \in \Omega$ and $r > 0$ we have
$$
u(x) = \frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)} u(y) dS(y).
$$
Now, the right hand side turns with transformation of variables into 
$$
\frac{1}{\omega_n r^{n-1}} \int_{\partial B_1} u(x+ry)dS(ry) = \frac{1}{\omega_n} \int_{\partial B_1} u(x+ry) dS(y).
$$ 
Now lets differentiate the right hand side with respect to $r$ (where by dominated convergence we can differentiate under the integral):
$$
\frac{d}{dr}\;\frac{1}{\omega_n} \int_{\partial B_1} u(x+ry) dS(y)
=
\frac{1}{\omega_n} \int_{\partial B_1} \frac{d}{dr}\;u(x+ry) dS(y)
=
\frac{1}{\omega_n} \int_{\partial B_1} (\nabla u)(x+ry)\cdot y\, dS(y)
$$
Now, changing the varibales back again, we get:
$$
\frac{1}{\omega_n} \int_{\partial B_1} (\nabla u)(x+ry)\cdot y\, dS(y)
=
\frac{1}{\omega_n r^{n-1}} \int_{\partial B_r(x)} \nabla u(x)\cdot \frac{(x-y)}{r}\, dS(y)
$$
which with the divergence theorem and the fact that $u$ is harmonic turns into
$$
\frac{d}{dr}\;\frac{1}{\omega_n} \int_{\partial B_1} u(x+ry) dS(y)
=
\frac{1}{\omega_n r^{n-1}} \int_{\partial B_r(x)} \Delta u dy = 0. 
$$
From this we can infer that the mean value representation actually is constant for $r$. So, it is left to show that the right hand side in the very first equation tends to $u(x)$ when radius $r$ shrinks (from above) to zero. But this is also not to difficult to see. 
Version 1b
I am aware that I didn't use your cited formula and to be honest I am a bit baffled by it. But isn't it just this argument: Suppose $u$ is harmonic then $\Delta u(x) \equiv 0$. Now, this means we have
$$
0 = \lim_{r\to 0} \frac{2n}{r^2}\left(\frac{1}{\omega_n} \int_{\partial B_1} u(x+ry)dS(y) - u(x)\right).
$$
But for this to be zero we need 
$$
\lim_{r\to 0} \frac{1}{r}\left(\frac{1}{\omega_n} \int_{\partial B_1} u(x+ry)dS(y) - u(x)\right) = 0
$$
and
$$
\lim_{r\to 0} \left(\frac{1}{\omega_n} \int_{\partial B_1} u(x+ry)dS(y) - u(x)\right) = 0
$$
to be zero and now that I typed it, I think that is actually the same argument I used :). Because that is the statement.
I would love some feedback !
