Lemma 14.21.8. Let $U \subset V$ be simplicial sets, with $U_ n, V_ n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then there exists a sequence of sub simplicial sets

$U = W^0 \subset W^1 \subset W^2 \subset \ldots W^ r = V$

such that Lemma 14.21.7 applies to each of the inclusions $W^ i \subset W^{i + 1}$.

Proof. Let $n$ be the smallest integer such that $V$ has a nondegenerate simplex that does not belong to $U$. Let $x \in V_ n$, $x\not\in U_ n$ be such a nondegenerate simplex. Let $W \subset V$ be the set of elements which are either in $U$, or are a (repeated) degeneracy of $x$ (in other words, are of the form $V(\varphi )(x)$ with $\varphi : [m] \to [n]$ surjective). It is easy to see that $W$ is a simplicial set. The inclusion $U \subset W$ satisfies the conditions of Lemma 14.21.7. Moreover the number of nondegenerate simplices of $V$ which are not contained in $W$ is exactly one less than the number of nondegenerate simplices of $V$ which are not contained in $U$. Hence we win by induction on this number. $\square$

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