Show by example that the hypothesis b1>=b2>>=0 cannot be replaced by bk>=0 and limit k-->infinity =0

hint: use |ab|< 1/2(a2+b2)

I've found an example: bk=(1/k2 + 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?