Most elegant way to proof that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it I was wondering which proof method can be used best to show that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it?

Consider
$$ \sum_{k=1}^n|x_n| = \left|\sum_{k=1}^n|x_n|\right| = \left(\left(\sum_{k=1}^n |x_n|\right)^2\right)^{1/2} \geq \left(\sum_{k=1}^n |x_n|^2\right)^{1/2} =  \left(\sum_{k=1}^n x_n^2\right)^{1/2}, $$
because $y^2 = |y|^2$. If you expand the expression for $\left(\sum_{k=1}^n |x_n|\right)^2$ you get all of $\sum_{k=1}^n |x_n|^2$ plus some additional nonnegative terms. Also, the square root function $t\mapsto t^{1/2}$ is monotonic increasing.
