Could someone help me prove the following:
If lim (an) = 0 and bn is a bounded sequence, prove lim (anbn) = 0.
Both limits are to infinity.
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