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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 $\lim_{n\to \infty}$ $\frac{a_n}{b_n}=L$.

Lord_Farin
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Vinny
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    A hint: consider an epsilon-delta definition of the first limit and ask what you can say about $\dfrac{a_{n+k}-a_n}{b_{n+k}-b_n}$ for sufficiently large $n$ and some fixed $k$ based on that; then 'flip it on its head' and find an epsilon-delta estimate for $\dfrac{a_{n+k}}{b_{n+k}}$ as $k\to\infty$ starting with some fixed $n$. – Steven Stadnicki Nov 15 '12 at 03:27
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    Please, try to make the titles of your questions more informative. E.g., Why does $a\le b$ imply $a+c\le b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. I've done so for you now. – Lord_Farin Aug 24 '13 at 08:22
  • @Lord_Farin You realize that the OP last (and probably first) visited the site on Nov 15 '12? – Did Aug 24 '13 at 09:28
  • @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. – Lord_Farin Aug 24 '13 at 09:36
  • Wikipedia: Stolz–Cesàro theorem. See also: limit of quotient of two series and Stolz-Cesàro Theorem. – Martin Sleziak Jun 15 '22 at 11:03

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

Glorfindel
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Mhenni Benghorbal
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