For each \(n \in \mathbb{N}\), let \(A_n = \left\{ 0,\dots,n \right\}\). What are the elements of \(\displaystyle\prod_{n\in\mathbb{N}} A_n\)?
By definition, \[\prod_{n\in\mathbb{N}} A_n = \left\{ f : f\text{ is a function } \land \operatorname{dom}(f) = \mathbb{N}\land \forall i\in\mathbb{N}\left[ f(i) \in A_i \right] \right\} \] which is the collection of all functions \(f : \mathbb{N}\to \mathbb{N}\) where for each \(i\in\mathbb{N}\), \(f(i) \in \left\{ 0,\dots,i \right\}\).
I think of it as the collection of all sequences bounded above by the sequence \(X_n = n\), that is, at each coordinate \(f(i) \leq X_i\).