Let the sequence {an} {BN} be Lim an BN = 0. If an is unbounded, then BN must be bounded. Is it correct? Why?

Let the sequence {an} {BN} be Lim an BN = 0. If an is unbounded, then BN must be bounded. Is it correct? Why?

Obviously wrong, for example
an = 1,0,2,0,3,0,4,0,...
bn = 0,1,0,2,0,3,0,4,...

Let an be an unbounded sequence and BN be an infinite sequence, and prove that an BN must be an unbounded sequence

Using the counter proof method: suppose an * BN is a bounded sequence, then by definition, there exists m > 0, for any n > 0, | an * BN | + ∞ (n - > + ∞), we know that there exists N2 > 0, when m > N2, | BM | > √ m (1) If an is an unbounded sequence, we know that there exists N1 > N2, such that | an1 | > √ m (2) In (1), let m = N1, (1) * (2): | a