Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, \;\left(a_{n} \neq 0\right) .$
Let $A=\max \left\{\left|a_{0}\right|,\left|a_{1}\right|, \ldots,\left|a_{n}\right|\right\}$.
Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, \left(a_{n} \neq 0\right) .$ and let $B=n A /\left|a_{n}\right|,$ Show that $f(x) \neq 0$ if $|x|>B .$
I have been trying to do this by induction and have successfully proved the base case but am unable to prove the induction step. Can someone point me in the right direction?
It is obvious that $B=nA/{a_n}\ge n\ge1$
For $|x|>B$, we have $\frac{|f(x)|}{a_n|x|^n}\ge|1|-|\frac{a_{n-1}}{a_n}x^{-1}|-...-|\frac{a_0}{a_n}x^{-n}|\gt 1-\frac B n (B^{-1}+B^{-2}+...+B^{-n})\ge1-\frac Bn(nB^{-1})=0$