Problem #PRU-61308

Problem

The numbers $a_1,a_2,\dots ,a_k$ are such that the equality $\underset{n\to \mathrm{\infty }}{lim}(x_n+a_1{x}_{n-1}+\cdots +a_k{x}_{n-k})=0$ is possible only for those sequences $\{x_n\}$ for which $\underset{n\to \mathrm{\infty }}{lim}{x}_n=0$. Prove that all the roots of the polynomial P $(\lambda )={\lambda }^k+a_1{\lambda }^{k-1}+a_2{\lambda }^{k-2}+\cdots +a_k$ are modulo less than 1.