strong maximum principle - harmonic function

strong maximum principle - harmonic function

Consider the following the theorem in the classical PDE book of Evans(
chapter 2 ) :
(part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u
\in C^2 (U) \cap C(\overline{U})$, with $\Delta u = 0$ in $U$.
If $U$ is connected and there exists a point $x_0 \in U$ such that
$$ u(x_0) = \displaystyle\max_{\overline{U}} u$$
then $u$ is constant within $U$.
Proof:
Suppose there exists a point $x_0 \in U$ with $u(x_0) = M =
\displaystyle\max_{\overline{U}} u . $ Then for $0 < r < dist (x_0 ,
\partial U)$, the mean value property asserts
$$ M = u(x_0) = \displaystyle\frac{\displaystyle\int_{B(x_0,r) } u \
dy}{|B(x_0, r)|} \leq M$$ .
Then $u = M$ in $B(x_0 , r)$ (*) . I dont understand the equality in
$(*)$, If i be non rigorous, for me is clear to see the equality in $(*)$.
But i dont know how to prove the equality... Someone can give me a hint
about how to prove the equality in $(*)$ ?
thanks in advance !