If you don't like the idea of fixing x, then simply replace $\displaystyle x_n$ by $\displaystyle f_n(x)$ and $\displaystyle y_n $ by $\displaystyle (f_n(x))_-$ throughout the proof. I was just trying to make it easier to read by simplifying the notation. Obviously I failed.

The point is that all these operations (limsup and liminf of a sequence of functions, taking the positive or negative part) are defined pointwise. So you might as well look restrict attention to what happens at a single point x.