How to prove that frequency mixer is a nonlinear system?
If the system is nonlinear then if $y_1(t)$ is the response to the signal $x_1(t)$, and  $y_2(t)$ is the output given input signal $x_2(t)$ then the response to the signal

$$x(t)=a_1x_1(t)+a_2x_2(t)\tag{1}$$

with arbitrary constants $a_1$ and $a_2$ will generally not be equal to

$$y(t)=a_1y_1(t)+a_2y_2(t)\tag{2}$$

However, for the given system an input signal (1) always results in an output signal (2). The conclusion is that the system is linear.

The confusion arises because you think that linear systems cannot add new frequencies to the input signal. This is generally not true. Only linear and time-invariant (LTI) systems have that property. Linear but time-varying systems can produce new frequencies in the output signal, and a modulator is a classic example of a linear time-varying system.