Let $X$ a simplicial set, $x\in X_m$ an $m$-simplex. We say that $x$ is degenerate if $\exists s:[m]\to[n]$ an epimorphism such that $n<m$ and $y\in X_n$ such that $X(s)(y)=x$.
Now I want to show Eilenberg-Zilber's lemma, that is : for each $x\in X_m$, $\exists s:[m]\to [n]$ an epimorphism and $y\in X_n$ non degenerate $n$-simplex such that $X(s)(y)=x$, and $(s,y)$ is unique.
The lectures I found or books (Calculus of fractions and homotopy theory, p.26/27) say that the existence is obvious, and then proceeds to show uniqueness. Obvious, sure. I thought about taking $n$ minimal in some sense but which sense ? And how to get this $y$? How can we be sure that it's non degenerate ?
In fact my question is simple, how is existence shown ?