As b_n is bounded there exists a b>0, such that |b_n|<b.

Also as lim_{n->infty} a_n=0, for any epsilon>0, there exits an N, such that

|a_n|<epsilon for all n>N.

Hence |a_n b_n|<epsilon b, for all n>N.

So for any epsilon', set epsilon = epsilon' /b, then there exists an N

such that |a_n| < epsilon, and also |a_n b_n|<epsilon' , hence:

lim_{n -> infty} a_n b_n = 0.

RonL