Analyzing a Closed Set on the Complex Line

[Somefantastik]

Homework Statement

on the complex line, with the usual metric, I need to determine if this is a closed set.

$$A = \left\{\left|\frac{1}{z^{2}+1} \right|: |z| = 1 ; z\neq \pm i\right \}$$

Homework Equations

The Attempt at a Solution

A closed set implies that the set of all limit points belongs to A.

Usually I'm given a function, and I take an arbitrary convergent sequence and show whether or not that point to which it converges is in A or not. But when I have just a set like this, I'm unsure of how to do that. Any advice?

 

[foxjwill]

Try finding a way to figure out what points are in the set.

Also, you might find it worthwhile to think of A as the image (your prof might call it the range) of the function $f\colon \{z:|z|=1, z\neq\pm i\}\to\mathbb{C}$ defined by

$$f(z) = \left|\frac{1}{z^2+1}\right|.$$​