The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidski(i) inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.