Moderate# Nets and Mobius Transformations

TOPO-9N3Y6$

What is the limit of the net $f:\mathbb C\cup\{\infty\}\rightarrow\mathbb C\cup\{\infty\}$ sending $z\mapsto (z-1)/(z+1)$, where $\mathbb C\cup\{\infty\}$ as a directed set has the partial order given by comparing absolute values?