Show by example that the hypothesis b_{1}>=b_{2}>=...b_{n}>=0 cannot be replaced by b_{k}>=0 and limit k-->infinity =0

hint: use |ab|< 1/2(a^{2}+b^{2})

I've found an example: b_{k}=(1/k^{2}+ 1/k) which satisfies the limit going to zer and all terms being positive, which diverges. I'm getting stuck with a rigorous proof of this, I thought about using the Dirichlet Test to prove this, but I'm getting hung up I think. Any help?