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Math Help - Another limit proof with divisibility of 2 sequences

  1. #1
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    Another limit proof with divisibility of 2 sequences

    Let (a_n)_{n \in \mathbb{N}} and (b_n)_{n \in \mathbb{N}} be sequences, both greater then 0. Assume that \frac{a_n}{b_n} \rightarrow X where 0<X<\infty. Show that a_n \rightarrow \infty if and only if b_n \rightarrow \infty. [Hint: eventually \frac{1}{2}X < \frac{a_n}{b_n} < \frac{3}{2}X ].

    I was originally thinking of using the definition: \left| \frac{a_n}{b_n} -X \right| <\epsilon but I realized that I was getting nowhere slowly. Looking at the hint I would think that \left| \frac{a_n}{b_n} -X \right| <\frac{1}{2}. At this point I don't know how to proceed.
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    \begin{gathered}<br />
  \frac{X}<br />
{2} > 0\; \Rightarrow \;\left( {\exists N} \right)\left[ {n \geqslant N\; \Rightarrow \;\left| {\frac{{a_n }}<br />
{{b_n }} - X} \right| < \frac{X}<br />
{2}} \right] \hfill \\<br />
   - \frac{X}<br />
{2} < \frac{{a_n }}<br />
{{b_n }} - X < \frac{X}<br />
{2} \hfill \\ <br />
\end{gathered}
    Now just add X.
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