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Suppose $a_1,\ldots,a_k$ and $b_1,\ldots,b_k$ are complex numbers bounded in absolute value by $1$. Is it true that $$ \left| \prod_{i=1}^k a_i - \prod_{i=1}^k b_i\right|\leq \sum_{i=1}^k |a_i-b_i|? $$

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Consider the telescoping sum $$a_1\cdots a_k-b_1\cdots b_k=\sum_{i=1}^k a_1\cdots a_{i-1}(a_i-b_i)b_{i+1}\cdots b_k. $$

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