Lemma 14.11.5. Let $U$, $V$ be simplicial sets. Let $a, b \geq 0$ be integers. Assume every $n$-simplex of $U$ is degenerate if $n > a$. Assume every $n$-simplex of $V$ is degenerate if $n > b$. Then every $n$-simplex of $U \times V$ is degenerate if $n > a + b$.

Proof. Suppose $n > a + b$. Let $(u, v) \in (U \times V)_ n = U_ n \times V_ n$. By assumption, there exists a $\alpha : [n] \to [a]$ and a $u' \in U_ a$ and a $\beta : [n] \to [b]$ and a $v' \in V_ b$ such that $u = U(\alpha )(u')$ and $v = V(\beta )(v')$. Because $n > a + b$, there exists an $0 \leq i \leq a + b$ such that $\alpha (i) = \alpha (i + 1)$ and $\beta (i) = \beta (i + 1)$. It follows immediately that $(u, v)$ is in the image of $s^{n - 1}_ i$. $\square$

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