Weak convergence of scaled elements implies norm convergence

Question

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , 2 <q \leq \infty$.

Progress

I am trying to show that $\sum_{n} u_{n}^{k} - n_{k}{\phi}_{n} \rightarrow 0$ but I can't. Here $\{ u_{n} : n \in \mathbb{Z} \}$ are the distributions and $\{{\phi}_{n} : n \in \mathbb{Z}\}$ are their respective characteristic functions.

Answer 1

Suggestions:

1. Recall that every weakly convergent sequence is bounded in the norm.
2. Think of what it means for $n_{k}u_{k}$ to be bounded in the norm for every sequence $n_k$
3. Conclude that your sequence $u_k$ is very special.