Let an > 0 and suppose that an converges. Construct a convergent series bn with bn

Question:

Let an > 0 and suppose that ∑an converges. Construct a convergent series ∑ bn with bn > 0 such that lim(an/bn)= 0; hence ∑bn converges less rapidly than ∑an. [Let (An) be the partial sums of ∑an and A its limit. Define b1 := √A - √A - A1 and bn := √A - An-1 - √A - An for n > 1.]