For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} \in \mathbb{R}, \,\, \|z\|^2=\sum_{\alpha \in \mathbb{Z}_{\geq 0}^n } z_{\alpha}^2 < \infty\}.$$

Let $1 \geq \epsilon>0$ be arbitrary small. Is the $\mathbb{R}$-span of the set $\{(x^\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}: x \in (0,\epsilon)^n\}$ dense in $\ell^2$? I would guess that should be known, does somebody know a reference?