Proof of linearity of a transistor
You cannot prove the linearity of a transistor, because a transistor is non-linear.

However, it's convenient to make linear models to simplify calculations about its operation. 

By definition, those linear models are linear. 

The way we make a linear model is to take the slope of the curve at a particular point, and use that as the 'linearised' response, around that point. This is a 'small signal' model, explicitly stating the fact that it's only valid for 'small' variations around that point.

Small signal models cease to be useful when we want to use them over parameter ranges where the non-linearity becomes apparent, that is, for large signals.