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?

Since you are dealing with a unit vector, $u$ each of the components $u_i$ must be at most one. Hence, for each $i$, $u_i^2 \le \lvert u_i\rvert$. So
$$
\lVert u\rVert_2^2 = \sum_i u_i^2 \le \sum_i \lvert u_i\rvert = \lVert u\rVert_1
$$
