There does not exist an elementary proof. Sadly, there is no proof at all, because the fact is **false**. Here is a construction of an $\ell^1$-space containing every $\ell^2_n$.

Consider the normalized Gaussian measure $d\mu=\pi^{-1/2}e^{-t^2}dt$. Form its infinite dimensional tensor product $d\mu_\infty$ over ${\mathbb R}^{\mathbb N}$. Now let $E$ denote the vector space of linear functions
$$f_a:x\longmapsto a\cdot x,\qquad x\in{\mathbb R}^{\mathbb N},$$
where $a$ has finite support. $E$ is contained in $L^1(d\mu_\infty)$. One sees easily, from the rotational invariance of $d\mu$, that
$$\|f_a\|_1\left(=\int|f_a(x)|d\mu_\infty(x)\right)=C\|a\|_2$$
where
$$C=\int_{\mathbb R}|t|d\mu(t).$$
Hence $L^1(d\mu_\infty)$ contains every $\ell^2_n$.

**Nota**. This argument is used to prove that Euclidian spaces satisfy the Hlawka Inequality. The latter is almost trivial in an $\ell^1$ space.