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 ?

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