Let $(a_k)_{k \in \mathbb{N}} \subset ]0,+\infty[$, and assume that $a_k \to 0$ as $k$ goes to $+\infty$, but $a_k \notin \ell^1(\mathbb{N})$.

It is easy to prove that we can extract a subsequence $(a_{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$; for this it is enough to choose $k_n$ going to $+\infty$ "fast enough".

I wonder if the following implication is true: $(a_{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$ $\Rightarrow$ $(\frac{1}{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$ ?

I also wonder if there is a way to reverse this implication, by doing an additional summability assumption like $(a_k)_{k \in \mathbb{N}} \in \ell^p(\mathbb{N})$?


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