Convergence of a series

if series a_n=A from and series b_n=B then series (a_n)(b_n)=AB on the interval of n=1 to infinity

is this statement True or False and why?

False.
Here is an example:
{a_n} = {1,0,1/2,0,1/4,0,1/8,0........}
{b_n} = {0,1,0,1/2,0,1/4.....}
Then A = B = 2 , but (a_n)(b_n) = 0 for every n.