how about
0<=(a-b)^2
2ab<=a^2+b^2
2ab<=a^2+b^2
therefore
2*sum(ab)<=sum(a^2)+sum(b^2)
Where the RHS converges
As the LHS converges absolutely sum(ab) converges?
Any ideas about the examples?
this is from a past paper by the way, I can link you if you like, it's not a take home test