Let $a_n$ be a diverging sequence and $b_n$ be a converging sequence. With that in mind does $\frac{a_n}{b_n}$ always diverge?
If so, what is the proof for this?
$\endgroup$ 11 Answer
$\begingroup$Let $b_n$ converge to $l_1$, and suppose $\frac{a_n}{b_n}$ converges to $l_2$. Then by the Algebraic Limit Theorem we have$$\lim_{n\to \infty}b_n\cdot \frac{a_n}{b_n} = l_1l_2.$$ But $$b_n \cdot \frac{a_n}{b_n} =a_n,$$ which is divergent, thus giving us a contradiction. Therefore $\frac{a_n}{b_n}$ is divergent.
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