Product of Convergent and Divergent Product is Divergent
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Theorem
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let $\ds \prod_{n \mathop = 1}^\infty a_n$ be convergent.
Let $\ds \prod_{n \mathop = 1}^\infty b_n$ be divergent.
Then $\ds \prod_{n \mathop = 1}^\infty a_n b_n$ is divergent.
Proof
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