if $\sum_{j\in\mathbb{Z}}|u_j|^p<\infty$, is then $u_j\to 0$?

Question

Let $u\in l^p(\mathbb{Z})$ with $1<p<\infty$ and $p$ is fixed. Per definition it is $\sum_{j\in\mathbb{Z}}|u_j|^p<\infty$. Does follow $u_j\to 0$ for $j\to \infty$?

I already know that $sup_j|u_j|<\infty$ and $|u_j|^p\to 0$. But I'm not sure how to conclude that $|u_j|\to 0$ for $j\to \infty$. Can you help me?