IMO 1970 Problem 3

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$ .

a.) Prove that $0\le b_n<2$ .

b.) Given $c$ satisfying $0\le c<2$ , prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$ .