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Suppose $(a_n)$ and $(b_n)$ are sequences where $b_n$ is increasing and approaching positive infinity. Assume that $\lim_{n\to \infty}$ $\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$, where $L$ is a real number. Prove that $\lim_{n\to \infty}$ $\frac{a_n}{b_n}=L$.

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    @Did Hm... I must have been put off by the fact that the question showed up on the main page. Oh well, it can't hurt to retain the comment, I suppose.2013-08-24

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This result is known as "Stolz-Cesaro theorem". See here for the proof.