 11D

The two preliminary sequence limit laws follow directly from the triangle inequality. For the sum, choose indices so that $|x_n-x|<\epsilon /2$ and $|y_n-y|<\epsilon /2$ beyond their respective cutoffs. Beyond the larger cutoff,Thus

For the reciprocal, convergence to nonzero $x$ gives $|x_n-x|<|x|/2$ eventually. The reverse triangle inequality then gives $|x_n|\ge |x|/2$, and consequentlyChoosing the eventual error smaller than both $|x|/2$ and $\epsilon |x{|}^2/2$ provesThe nonzero limit gives the denominator a positive lower bound; nonzero individual terms alone would not do so. The assumption $x_n\ne 0$ ensures the reciprocal sequence is defined even before this eventual bound applies.