electrostatics - Confusion in understanding the proof of Uniqueness Theorem

I am having problems in comprehending the proof of contradiction used by Purcell in his book;

...We can now assert that $W^1$ must be zero at all points in space. For if it is not, it must have a maximum or minimum somewhere-remember that $W$ is zero at infinity as well as on all the conducting boundaries. If $W$ has an extremum at some point $P$, consider a sphere centered on that point. As we saw in chapter2, the average over a sphere of a function that satisfies Laplace's equation is equal to its value at the center. This could not be true if the center is extremum; it must therefore be zero everywhere. $^1 W = \phi(x,y,z) - \psi(x,y,z)$, where the former term is the deduced solution & the later term is the assumed solution in order to proof contradiction.

Extremum means local maximum or local minimum, right? Why can't average be equal to an extrmum value? If it is not equal to the average value, how does it ensure that at all the places $W$ is equal?

Answer

Well, if you have an extremum (say local maximum) of $W$ at $p$, then you have a small open ball $N$ centered in $p$ such as $W(x)< W(p)$ for some $x\ne p$ in $N$. Therefore $$ \frac{1}{{\rm vol}(N)}\int_N W(x)dx < W(p)~~.\tag{1}$$ assuming we have taken the ball $N$ small enough for $W(x)