Examples of Axiom of Choice used in introductory-level undergradute math
The Axiom of Choice can be used to prove that every vector space has a basis. That's certainly a basic result from a linear algebra course but it's definitely not unbeknownst to students.
The axiom of choice can be used to prove that the sequential definition of continuity at a point (for real functions of a real variable) is equivalent to the $\varepsilon$-$\delta$ definition. If your calculus textbook proves that sequential continuity implies epsilon-delta continuity without mentioning the axiom of choice, it's doing something like this:
. . . Then, for each $n\in\mathbb N,$ there is a real number $x_n$ such that $|x_n-x_0|\lt\frac1n$ while $|f(x_n)-f(x_0)|\ge\varepsilon.$ Thus the sequence $x_n$ converges to $x_0,$ while $f(x_n)$ does not converge to $f(x_0)$. . .
Do you see where I used the axiom of choice?
Here are two which you usually see in the first chapter of most intro to analysis books, and in many "quick set theory coverage" of the first couple of weeks:
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Every infinite set has a countably infinite subset. Or equivalently, a set is finite if and only if every proper subset has smaller cardinality. Or equivalently, a set is infinite if and only if there is an injection which is not a surjection from the set to itself.
You need strictly less than countable choice to prove this, even though the standard and easy proof is using a bit more. But this is certainly not provable without any use of choice. Not even if these are sets of reals.
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The countable union of countable sets is countable. Well, even the real numbers can be a countable union of countable sets if choice fails badly enough.
If it is up to me, I'd also include a third, but I am not sure if fits your criteria:
- A function is surjective if and only if it has a right inverse. This one is in fact equivalent to full choice, and it is often used in basic level courses like discrete mathematics and stuff like that. And it makes sense, but for infinite sets you generally still need choice.
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.