Why is the electric field inside a conductor zero?

This is actually a tricky question that is best answered not by showing how it works but by showing how it DOESN'T work. Here's my sort of round-about approach: I'll start with a definition, then use a counter argument to show a contradiction, thus proving the definition. Got it?

Definition:

When there is no net motion of charge within a conducting sphere, the conductor is in electrostatic equilibrium.

If a conductor is in electrostatic equilibrium, the free electrons on the surface of the conductor are not accelerating away from each other.

Contradiction:

If there WERE an electric field inside the conductor, the field would exert a force on the free electrons on the surface of the conducting sphere, which would cause them to accelerate.

Since the electrons in a conductor in electrostatic equilibrium are NOT moving away from each other, there can be no electric field inside the container.

Another question might involve the application of an electric field OUTSIDE the shere.

If this were to occur, say an external positive charge were placed in close proximity to the sphere, the electrons on the surface of the sphere would rearrange themselves, fleeing the positive charge and setting up an new electric field inside the container FOR JUST AN INSTANT. This new field would then cancel the external electric field.

As a result, after this nano-second, there would be no electric field present in the sphere.