# Division of two series

Let $y_n >0$ for all $n \in \mathbb{N}$ where $\sum {y_n} = + \infty$ and a sequence $(x_n)$ of real numbers. If $\lim\limits_{n \rightarrow \infty} \dfrac{x_n}{y_n} = a$ then $\lim\limits_{n \rightarrow \infty} \dfrac{x_1 + \dotsb +x_n}{y_1 + \dotsb + y_n} = a$.

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Hint: given $\epsilon > 0$, for $n$ sufficiently large $(a - \epsilon) y_n < x_n < (a+\epsilon) y_n$.
Suppose those inequalities are true for $n > N$, and let $x_1 + \ldots + x_N = A$. What inequalities does that give you for $x_1 + \ldots + x_n$? –  Robert Israel Oct 16 '12 at 18:40