How do I prove that $y(t) = x(t)x(t-1)$ is a non linear system?
I tried the following proof but it seems not to getting the desired effect. Most likely I solve it wrongly.
Superposition property:
$$y(t)=T[ x(t) ] = T[ X_1(t) + X_2(t) ] = X_1(t)X_1(t-1) + X_2(t)X_2(t-1)=y_1(t)+y_2(t)$$
Scaling property:
$$y(t)=T[ c x(t) ] = c T [x(t)]$$
Simply note that
$$y(t)=T[c\cdot x(t)]=c\cdot x(t)\cdot c \cdot x(t-1)=c^2\cdot x(t)\cdot x(t-1)=c^2\cdot T[x(t)]$$