Let $A$ be Von Neumann algebra.
A positive linear functional $\varphi$ on $A$ is said to be normal if for any selfadjoint and increasing nets such that $u_{\alpha}\longrightarrow u$ we have $\varphi(u_{\alpha})\longrightarrow \varphi (u)$.
I know that if the positive functional $\varphi$ is $\sigma$-weakly continuous then $\varphi$ is normal. I know that the converse holds but I don't know why.
Q: If $\varphi$ is normal then why is $\varphi$ $\sigma$-weakly continuous?