Examples of Axiom of Choice used in introductory-level undergradute math All of the applications of AoC I've encountered have been in upper level undergraduate or graduate math courses. Are there any basic results from courses like Calc I-III which (unbeknownst to students) rely on AoC? 

A bounded, closed, convex subset of $\Bbb{R}^n$ has extreme points.
Amazingly we need (a weaker version of) axiom of choice to prove this simple fact and this has many applications. For example:  To solve a linear program problem only needs to consider extreme points of its feasible region.
