It is proved that because {xn} is bounded, there exists a positive number m, which makes | xn | < m.

And because lim (n->; ∞)yn=0, so for any e >；; 0, there is a positive integer n, so that all n >；; N, you have | yn | < e

So |(xnyn)/M-0|=|xnyn|/M

=|xn|*|yn|/M

& ltM*e/M

=e

So lim (n->; ∞)xnyn/M=0

Because m is a constant, lim (n->; ∞)xnyn=0