Magnitude of a root of a polynomial

Assume the inequality holds for any root of any complex polynomial of degree at least $1$.

Then any root $\lambda$ of $P(X):=\sum_{i=0}^{n}a_{i}X^{i}$ is also a root of $\alpha P$ where $\alpha$ is a non-nul complex number. Hence if $\lambda$ is a solution of the inequality for $P$ then it has to be a solution for $\alpha P$. Hence, for all $\alpha\in\mathbb{C}^{*}$:
$$\vert\lambda\vert \leqslant \vert\alpha\vert\,\sum_{i=0}^{n}\vert a_{i}\vert$$But then one can let $\alpha$ goes to $0$ so $\lambda$ has to be zero. But there exists polynomials which roots are not zero thus this is our contradiction.