Question 3
Let $1=a_0\le a_1\le a_2\le\cdots\le a_n\le\cdots$ be a sequence of real numbers. Consider the sequence $b_1,b_2,\ldots$ defined by
$$b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}.$$
Prove that:
(a) For all natural numbers $n$, $0\le b_n\lt 2$.
(b) Given an arbitrary $0\le b\lt 2$, there is a sequence $a_0,a_1,\ldots,a_n,\ldots$ of the above type such that $b_n\gt b$ is true for an infinity of natural numbers $n$.
$$b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}.$$
Prove that:
(a) For all natural numbers $n$, $0\le b_n\lt 2$.
(b) Given an arbitrary $0\le b\lt 2$, there is a sequence $a_0,a_1,\ldots,a_n,\ldots$ of the above type such that $b_n\gt b$ is true for an infinity of natural numbers $n$.