I am reading Folland's real analysis. In Prologue, proof of Axiom of choice using well ordering is as follows:
Let $X=\cup_{\alpha \in A}X_{\alpha}$. Pick a well ordering on $X$ and, for $\alpha \in A$, let $f(\alpha)$ be the minimal element of $X_{\alpha}$. Then $f\in \prod_{\alpha \in A}X_{\alpha}$.
My question is that why it is necessary to pick the minimal element of $X_{\alpha}$ so that we would need well ordering principle. Why can't we just choose an arbitrary element of $X_{\alpha}$ while we know that $X_{\alpha}$ is non-empty so it has at least one element.
I think I have not understood the axiom of choice well.
You can choose an arbitrary element of one $X_\alpha$ without using the axiom of choose. For that matter, you can choose one arbitrary element from any finite number of the sets $X_\alpha$ without using the axiom of choice. But to choose one member from each of an infinite family of sets you need either the axiom of choice or a specific algorithm for choosing them. Here the well-ordering on $X$ provides that algorithm: it allows to choose from $X_\alpha$ very specifically the minimal element of $X_\alpha$ with respect to the well-ordering of $X$. This is no longer an arbitrary choice: it is completely determined by the well-ordering.