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I have the following proof of the sun rule for sequences : We have the two sequences $(a_n)$ and $(b_n)$ where $(a_n)\rightarrow a $ and $(b_n)\rightarrow b $. We also observe that $$0\leq |c(a_n-a)+d(b_n-b)|\leq |c||a_n-a|+|d||b_n-b| .$$ We know that both $(a_n-a) $ and $(b_n-b)$ are null sequences so using sum rule and sandwich theorem for null sequences (take as given here) we conclude that $(c(a_n-a)+d(b_n-b))\rightarrow 0 $ so $$(ca_n+db_n-(ac+bd))\rightarrow 0 \ \Rightarrow (ca_n+db_n) \rightarrow ac+bd , $$ since $(x_n-x)\rightarrow 0 \iff (x_n)\rightarrow x. \ \ \square $

Is this a valid proof or are there holes?

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