Let $(a_n)$ be a summable sequence of positive real numbers then we can find a sequence $w_n\to \infty$ such that the sequence $(a_nw_n)$ is still summable. This property has been asked and answered already. But I do not know if we can choose $(w_n)$ uniformly for a sequence $(a^k_n)$ convergent to $(a_n)$ in $l^1$ as $k\to \infty$.

  • $\begingroup$ This is actually not an exercise (homework) but emerges when I study partial differential equations. An element in $L^p$, Sobolev, Besov space...can be defined as a sequence in $l^r$ (for some suitable $r$). So if we are given data in these spaces, by using this problem we can have something better with the presence of the "weight" $w$. $\endgroup$ – user117042 Jan 7 '14 at 19:44

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