$\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$.

Question

Let $(H_{\lambda})_{\lambda\in \Lambda}$ is a family of Hilbert spaces and $A_{\lambda}$ is a von Neumann algebra on on $H_\lambda$ for each index $\lambda$. Then prove that the direct sum $\bigoplus_\lambda A_\lambda$ is a von Neumann algebra on $\bigoplus_\lambda H_\lambda$.
Basically we want to show that $\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$. Now notice that, by definition we have, $$\bigoplus_\lambda A_\lambda:=\left\{(a_\lambda)_\lambda : \sup_\lambda \|a_\lambda \|<\infty \right\}.$$ Now to show that $\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$, let $(u_\alpha)_\alpha$ be a net strongly converging to $u$, where $u_\alpha \in \bigoplus_\lambda A_\lambda$ for each $\alpha$. I want to show that $u\in \bigoplus_\lambda A_\lambda$. I know by double commutant theorem that if I show $\bigoplus_\lambda A_\lambda=\left(\bigoplus_\lambda A_\lambda\right)''$, then we are done. But I do not want to prove using double commutant theorem, I want to use definition of strongly closed. Please help me to solve that $\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$. Thanks.