The equivalent definition of singular poaitive linear functional on a von Neuamann algebra

Question

The screenshot is from Takesaki's book.

In Theorem 3.8. Suppose the condition in (ii) is satisfied, how to get the contadiction by uaing Lemma 3.6?

By Lemma 3.6, we only know that $\omega_n=s(\omega_n)\omega_n$. If $e=s(\omega_n)\geq e_0$, we have $ee_0=e_0$.

Answer 1

We also have that $\omega_n$ is faithful on $eAe$ by Lemma 3.6, and that $e_0=ee_0e\in eAe$. Since $e_0$ is nonzero, it then follows that $\omega_n(e_0)>0$. However, $$0=\omega(e_0)\geq\omega_n(e_0)>0,$$ a contradiction.