Proving that product of two Cauchy sequences is Cauchy
HINT: You can make the basic idea work by arranging matters a bit differently. Start with
$$x_ny_n-x_my_m=(x_ny_n-x_ny_m)+(x_ny_m-x_my_m)\;,$$
supply and manipulate absolute values appropriately, and use the fact that a Cauchy sequence is bounded. (Note that boundedness can be proved easily without actually showing convergence.)