On P145 of Murphy's book C$^*$-Algebras and Operator Theory, when he proves a character $\tau$ of an abelian C* algebra has norm one, he uses that $(u_\lambda^2)$ is also an approximate unit when $(u_\lambda) $ is an approximate unit.
But why? An approximate unit of a C* algebra $A$ is an increasing net of positive elements in the closed unit ball such that $u_\lambda a\to a\forall a \in A$. Note that $0\leq u\leq v$ dose not imply that $u^2\leq v^2$. Why is $(u_\lambda^2)$ increasing?