Question

When is sum of product of two sequences equal to 0?

Answer 1

Let an,bn⊂R be two sequences, where liman=a∈R a and limbn doesn't exist. What can we say about

lim(an+bn)

and

lim(an⋅bn)

I think that lim(an+bn) doesn't exist and lim(an⋅bn) has limit (0) only if liman=0, but I don't know how to prove it. (how to get those solutions)