Continuity of linear functionals

Question

We know from Operator Theory the following Theorem: A linear functional $f:A\rightarrow\mathbb{C}$ on a van Neumann algebra $A$ on an Hilbert space $H$ is ultra-weak continuous iff there is an trace class operator $p$ on $H$ such that $f(a)=tr(pa)$ for all $a\in A$. The claim is want to prove is the following: If $f$ is ultra-weak continuous, then there holds that for every increasing net $(v_i)_{i\in I}$ in the set $A_{sa}$ of self-adjoint operators which converges strongly to some $v\in A_{sa}$ we have $\lim_{i}f(v_i)=f(v)$. The hint i get is the following: Let $(u_i)_i$ a bounded net which converges strongly to some $u$ and if $p$ is a trace class operator then $\lim_i{\left\|u_ip-up\right\|_1}=0$ where $\left\|\cdot\right\|_1$ notates the trace class norm. I think i know why we have to follow the hint because we have Vigier's Theorem for nets of self-adjoint operators and we combine this to prove the fact we want, but i can not prove the hint. Can someone help me with this? Thank you so much.