Given $f,g$ holomorphic and $C$ a closed contour if $|g(z)|< |f(z)|$ on $C$ then $f$ and $f+g$ have the same number of zeros (counting multiplicity) in the interior of $K$. C$. There's an easy proof of this using the Cauchy integral formula. If Let$g(z) = a_{n-1} z^{n-1} + \cdots + a_0$, and$f(z) = z^n$. If$R$is sufficiently big then$|g(z)|<|f(z)|$on the circle of radius$R$with the center at 0. Thus$p(z) := z^n + g(z)$has$n$zeros inside that circle. [As a side note, when I was taught this by Lipman Bers, he picturesquely referred to it as the "dog on the leash theorem" -- it's essentially a winding number argument] 1 [made Community Wiki] I'm surprised that no-one's mentioned the proof using Roueche's theorem: Given$f,g$holomorphic and$C$a closed contour if$|g(z)|< |f(z)|$on$C$then$f$and$f+g$have the same number of zeros (counting multiplicity) in the interior of$K$. There's an easy proof of this using the Cauchy integral formula. If Let$g(z) = a_{n-1} z^{n-1} + \cdots + a_0$, and$f(z) = z^n$. If$R$is sufficiently big then$|g(z)|<|f(z)|$on the circle of radius$R$with the center at 0. Thus$p(z) := z^n + g(z)$has$n\$ zeros inside that circle.