Common Emitter amplifier is used as
Common collector (emitter follower) is a pure current amplifier. In fact, it has a tiny loss in terms of voltage.

Common base is a pure voltage amplifier. It has a tiny loss in terms of current.

Common emitter has both voltage and current gain in most configurations, and so is best characterized as a power amplifier.

I don't know what would constitute "proof" ... it's just a matter of evaluating any given circuit for both kinds of gain.