Let $A=\{a_n|n\in\mathbb{N}\}$ be a sequence of positive integers with the following properties:
$1<a_1<a_2<\cdots<a_n<\cdots$,
$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}+\cdots=\infty$.
Does there always exist a subsequence $B\subseteq A$ with $B=\{b_n|n\in\mathbb{N}\}$ such that
$\frac{1}{b_1}+\frac{1}{b_2}+\cdots+\frac{1}{b_n}+\cdots=\infty$,
All products $d_1^{m_1}\cdot d_2^{m_2}\cdots\cdot d_k^{m_k}$ are distinct (for all $k\in \mathbb{N}$), where $d_1,\dots,d_k$ are distinct elements of $B$ and $m_1,\dots,m_k$ are positive integers ?