수열의 귀납적 정의

Let $P(n)$ be $a_n = n!$. Since $a_0 = 1$ and $0 ! = 1$, we see that $P(0)$ is true. For the inductive step, we assume that $k \in \mathbb{N} \cup \{0\}$ and that $P(k)$ is true or that $a_k = k!$.

$$\begin{aligned}a_{k+1} & = \left( k + 1 \right)a_k \\ & = \left( k + 1 \right) k! \\ & = \left( k + 1 \right)!\end{aligned}$$

. This proves the inductive step that if $P(k)$ is true, then $P(k+1)$ is true.