For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$
But using Cauchy-Schwarz inequality, I get a "wrong" comparison:
$||x||_1=\sum_i|x_i|\leq\left(\sum_i|x_i|^2\right)^{\frac{1}{2}}\left(\sum_i 1\right)^{\frac{1}{2}}=\left(\sum_i|x_i|^2\right)^{\frac{1}{2}}=||x||_2.$
What is going wrong here? Thanks.