Is this proof for the limit law of the product of converging sequences correct? [duplicate]
Solution 1:
The goal is to make $|x_ny_n-xy|$ smaller than $\epsilon,$ and the trick is to use the triangle inequality: $$ |x_ny_n-xy|\leq |y_n-y||x_n|+|x_n-x||y|.$$ Since $|x_n|\leq B$ for all $n,$ they make $|y_n-y|<\frac{\epsilon}{2B}$ so that $|y_n-y||x_n|<\frac{\epsilon}{2},$ and $|x_n-x|<\frac{\epsilon}{2(|y|+1)}$ so that, since $\frac{|y|}{|y|+1}<1,$ we have $|x_n-x||y|<\frac{\epsilon}{2}.$ Then $$ |x_ny_n-xy|\leq |y_n-y||x_n|+|x_n-x||y|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$$ as desired.