We want to choose a distinct sequence of elements from an infinite set.1
This sometimes called the axiom of choice. It is equivalent to saying that if for each $\alpha \in I$ we can choose a point $x_{\alpha } \in X_{\alpha }$ then we may construct a function $x \in \prod_{\alpha \in I}X_{\alpha }$ by setting $x(\alpha ) = x_{\alpha }$