Let $p\in[1,\infty)$ and consider $\ell_p$. Let $A=\{x=(x_n)\in\ell_p: x_n\ge0\text{ for all }n\in \mathbb{N}\}$.
Is there a sequence $u=(u_n)\in\ell_p$ such that $\inf\{x, n\cdot u\}\uparrow x$ as $n\to\infty$ for all $x=(x_n)\in A?$ Here $n\cdot u$ denotes the pointwise product that is, $n\cdot u=(n\cdot u_1, n\cdot u_2,\ldots)$
I feel like such a $u=(u_n)$ cannot exists, because given an $\ell_p$ sequence, I can always find a sequence that converges at a faster rate. But I don't know how to process this into a correct mathematical argument.