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$$\left\|\sum A_iv_i\right\|=\sup_{\|x\|=1} \left\|\sum A_iv_ix\right\|\leq \sup_{\|x_i\|=v_i} \left\|\sum A_ix_i\right\|\leq \sup_{\|x_i\|=v_i} \|A\| \left\|\begin{pmatrix} x_1\\ \vdots\end{pmatrix}\right\|= \|A\| \|v\|.$$