I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself.
My question is the following. The image of a circumference of radius $r$ around the origin ($|z| = r$) through a complex continuous function $f(z)$ is a closed curve. If given an annulus such as that the image of the outer circumference completely contains the image of the inner circumference, how does that imply surjectivity in the region between those images?
That's what I want to have so I can formalize the notion of "shrinking" the domain so that the image crosses the origin (just how it's done in the sketch), something similar to the intermediate value theorem on $\mathbb{R}$. I had the following idea: given a ray on the annulus, it's image will be a simple curve with endpoints on the image of the outer and inner curves. By "sweeping" the ray around the annulus, the image curve will cover all the interior of the annulus. I don't know how to formalize it though.