Showing $(a_k)_{k=1}^{\infty}$ is in $l^1$ My lecturer claimed this following without a proof. If anybody could give me some idea why this is true, it would be great.
Let $a_k\in\mathbb{R}$, be a real sequence, such that the series $\sum_{k=1}^{\infty}a_kx_k$ converges for all sequences $(x_k)_{k=1}^{\infty}$ with $\lim_{k\rightarrow\infty}x_k=0.$ Then $(a_k)_{k=1}^{\infty}\in l^1$?

It suffices to show that $(a_n)_{n=1}^\infty\in (c_0)^*$. To show this, observe that the operators $x\mapsto \sum_{n=1}^N a(n)x(n)$ are continuous on $c_0$. And since for each fixed $x\in c_0$, the quantity $$\sup_N |\sum_{n=1}^N a(n)x(n)|$$ is bounded. So by the Uniform Boundedness Theorem, $a$ is continuous on $c_0$.
