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Following the other question: If a function is known to be non-neligible by this definition, (for example $q(x)=1/x$, is it true (provable) that $poly(x)*q(x)$ (for any positive polynomial function) is also a non-negligible function?

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Yes, trivially. If $poly(x)$ is a positive polynomial, then $poly(x) * q(x) > q(x)$ for all $x$. Since $q(x)$ is non-negligible, $poly(x) * q(x)$ is also non-negligible.

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