Stolz-Cesaro theorem

Let $(a_{n})_{n\\geq 1}$ and $(b_{n})_{n\\geq 1}$ be two sequences of real numbers. If $b_{n}$ is positive, strictly increasing and unbounded and the following limit exists:

\\lim_{n\\rightarrow\\infty}\\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}=l

Then the limit:

\\lim_{n\\rightarrow\\infty}\\frac{a_{n}}{b_{n}}

also exists and it is equal to $l$ .

Remark. This theorem is also valid if $a_{n}$ and $b_{n}$ are monotone, tending to $0$ .

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