I think you can remove the condition that at least one of the sequences x_1, x_2, ..., x_n and y_1, y_2, ..., y_n consists of positive numbers (I didn't use it in my proof, but it might be wrong). Also, IIRC x_1, x_2, ..., x_n isn't really a sequence (sequences consist of infinitely many numbers x_1, x_2, ...).
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