given that we have 2 series a_n and b_n that converges, does that mean that the product of this two series a_n b_n also converges?

what i said was that since both converge, the seq of finite partial sums for both will be bounded and hence

using cauchy seq,

sum from k = n to m of a_n b_n is also bounded.

since the product is cauchy, it is convergent.

is my reasoning right?

thanks