Finiteness of the Supremum of Inner Product of Two Finite Sum Positive Sequence

Question

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\limits_{i\in \mathbf N} a_ib_i =x\bigg\}\right).$$ Certainly $v$ is an increasing function. Is $v(x)$ finite for every $x$? Is it achievable? Does $v(x)\rightarrow 0$ as $x\rightarrow 0$?

Answer 1

\begin{align} \Big( \sum_i a_ib_i\Big)^2 &= \Big(\sum_i (a_ib_i)^\frac{1}{2}a_i^\frac{1}{2}b_i^\frac{1}{2}\Big)^2 \\ &\le \sup\limits_i (a_ib_i)\Big(\sum_i a_i^\frac{1}{2}b_i^\frac{1}{2}\Big)^2 \\ &\le \sup\limits_i (a_ib_i)\sum_i a_i\sum_ib_i. \end{align}