I'm reading the proof of Banach Alaoglu using nets. The theorem states that the closed unit ball in $X^\ast$ is weak star compact.
My question is: If $\varphi_\alpha$ is a universal net mapping into the closed unit ball why is $\varphi_\alpha (x)$ also universal?
A set is called universal iff for any set $Y$ it is eventually in $Y$ or in $Y^c$. The net $\varphi_\alpha(x)$ is a map $A \to \mathbb C$ (if $X$ is acomplex normed vector space).