Is there an intuitive argument to this? I know that the product of 2 Cauchy sequences is Cauchy, but that doesn't necessarily mean that all Cauchy sequences involve the product of 2 individual Cauchy sequences, does it?
I thought of the following:
The product of a "Non-Cauchy" sequence and a Cauchy sequence is always "Non-Cauchy" because successive terms would never get arbitrarily close to one another as their index increases, marked by the product of the divergent term. Is this logic bad?