I attached a picture with my confusion.
Hopefully, someone will be able to shed some light upon this.

 

When you claim

f(u[n] - u[n-4]) = f(u[n]) - f(u[n-4])

you are claiming that f() is a linear function.

Is it?

Why are you saying that

f(u[n-4]) = (n-4)u[n-4]

You have

f(x[n]) = n * x[n]

Whatever you plug in as the argument of f(), that only replaces the parameter x[n] in the expression.

 

Just a quick comment that the step f(u[n-4])=(n-1) u[n-4] is the incorrect step as mentioned above.

The step f(u[n] - u[n-4]) = f(u[n]) - f(u[n-4]) is correct because this is a linear time varying system.

 

I figured it out. I was confused between the actual model and the input-output relationship that arises from it.
Thanks for the answers.