Confusion regarding proof of Uniqueness theorem as explained in Purcell In Purcell it was written while showing that laplace equation has a unique solution :

We can now assert that if $W$ is zero on all the conductors, then $W$ must be zero at all points in space. For if it is not, it must have either a maximum or a 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 Section $2.12$, 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 a maximum or minimum. Thus $W$ cannot have a maximum or minimum; $^{4}$ it must therefore be zero everywhere. It follows that $\psi=\phi$ everywhere, that is, there can be only one solution of Eq. (3.1) that satisfies the prescribed boundary conditions.


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*My query is as such $W$ is zero at some finite distances in the space on the surfacw of conductors and also zero at $\infty$ but why does that means it has a $maxima /minima$ in between as such doesnt a function in  a closed interval has a maxima/minima then we can apply "extreme value theorem" ? Here the interval is not closed isnt ? So how are we sure of that ? Asssuming the laplace function here would be continuous .  One more query related to same , it was just above the uniqueness theorem 3.1


We shall prove that this boundary-value problem has no more than one solution. It seems obvious, as a matter of physics, that it has a solution, for if we should actually arrange the conductors in the prescribed manner, connecting them by infinitesimal wires to the proper potentials, the system would have to settle down in some state.


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*What does it means to say when saying connecting them by infiniteismal wires to the proper potentials , it should definitely come to settle come down in some state? Why should it be?

 A: He simply saying if you draw any continuous curve that is zero at two
distinct points, if it is nonzero between the points it must have a
maximum or a minimum there.  Just draw some curves if this isn't obvious.
To be pedantic, to apply this, draw any path between your conductors
and infinity. Along this path it will have a maximum or minimum or
be everywhere zero. Repeat for paths so that at least one path passes
through all points. You will find either all paths give zero or there
is a maximum or minimum in three-dimensional space.  Solutions to
Laplace's equation cannot have maxima or minima (except at bounaries)
because the sum of the second derivatives must be zero. This sum is
positive or negative for minima and maxima. Purcell uses the mean value
theorem instead for his reasoning.
For your second question, he is asking you to be a physicist and imagine
doing the experiment. Take your conductors say of size 1 meter or so in
your space. You then take wire with a diameter of say 0.1 millimeter and
connect the conductors with batteries to bring them to their desired
potentials. You can then remove the wires and batteries if you desire
and measure the electric field.  Since the wire is much smaller than the
conductors, the charge on the wire is a small change to the system. He
is then telling you to make the wires as small as necessary to make
their effect on the potential negligible. His point is that you can
experimentally force the conductors to have the desired potentials, and
this has some measurable field. It is obvious to him, and he thinks it
will be obvious to you, that it will be the same every time you set up
this experiment.
