If algebraic manipulations fail, I find it useful to think intuitively about what happens at large n. The idea is that x_i will be become sufficiently close to x, so, roughly speaking, they can be factored out.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.

Let . Then . Suppose is given and we need to make this fraction less than (in absolute value) by choosing a sufficiently large n.

We represent the fraction as a sum where is chosen so that for . Then the second term, regardless of , is less than . For this given , the numerator of the first term is fixed, and since the denominator diverges, one can choose so that the first term is also less than .