Assume that $(m_{\alpha})_{\alpha \in \Lambda}$ is a bounded net in Banach algebra $A$ and $(n_{\gamma})_{\gamma\in \Gamma}$ is a bounded net in Banach algebra $B$. Let $(t_{(\alpha,\gamma)})_{(\alpha,\gamma)\in\Lambda\times\Gamma}=((m_{\alpha}),(n_{\gamma}))$ be a net in $A\oplus_{\infty} B$. Is the following relation true?
$\lim_{(\alpha,\gamma)}(t_{(\alpha,\gamma)})=\lim_{(\alpha,\gamma)}((m_{\alpha}),(n_{\gamma}))=(\lim_{\alpha}(m_{\alpha}),\lim_{\gamma}(n_{\gamma}))$