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Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case that $(a_n,b_n)$ converges to $(0,0)$ (I strongly believe so), and how can this be shown?

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