Let $(\lambda_j)$ be a monotonically decreasing sequence of positive numbers converging to zero but so that $\sum_{j=1}^\infty\lambda_j=\infty$, and let $(u_j)$ be an arbitrary sequence of non-negative real numbers. Assume that $$ \sup_\sigma\sum_{j=1}^\infty\lambda_{\sigma(j)}u_j<\infty, $$ where the supremum is taken over all permutations $\sigma$ of $\mathbb N$.
Question: Is it always true that $$ \lim_{n\to\infty}\left(\sup_\sigma\sum_{j=1}^\infty\lambda_{\sigma(j)}u_{j+n}\right)=0? $$
I really have no idea if this is true or not. Note that $$ \sup_\sigma\sum_{j=1}^\infty\lambda_{\sigma(j)}u_{j+n}=\sup_\sigma\sum_{j=n+1}^\infty\lambda_{\sigma(j-n)}u_{j} $$ and so the whole thing reminds me of the fact that the remainder of a convergent series goes to 0. However, I do not know how to handle the shifting that is done by the permutation. Any ideas are highly appreciated. Thanks in advance!