There exists a sequence $\displaystyle (x_{n})$ such that $\displaystyle \lim_{n \longrightarrow \infty} x_{n} = x$ and there exists a sequence $\displaystyle (y_{n})$ of non-negative real numbers such that $\displaystyle \lim_{n \longrightarrow \infty} y_{1} + y_{2} + y_{3} + ... + y_{n-1} + y_{n} = \infty$ .

Show: $\displaystyle \lim_{n \longrightarrow \infty} \frac{x_{1}y_{1} + x_{2}y_{2} + ... + x_{n-1}y_{n-1} + x_{n}y_{n}}{y_{1} + y_{2} + ... + y_{n-1} + y_{n}} = x$ .

I'm having some trouble figuring out where to begin on this one. I've tried re-writing it in summation notation and manipulating that in hopes of being able to cancel y terms but hasn't gotten me anywhere. I thought it might be possible to rewrite the expression as a subsequence of $\displaystyle (x_{n})$ and then take the fact that a subsequence converges to the same limit as the original sequence but I would not no where to begin to construct the subsequence.