Let $(a_n)$ and $(b_n)$ be sequences of real number. Let $(\frac{a_n}{b_n})$ and $(a_n b_n)$ are convergent, then
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To test general statements in analysis questions, try using the sequence \(\mathbf{(-1)^n}\) as a quick test case.
Its oscillating behavior makes it an excellent source of counterexamples for questions about boundedness and convergence.