Equivalence of Definitions of Factorial

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $\begin {cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end {cases} \equiv \ds \prod_{k \mathop = 1}^n k$

Basis for the Induction
$\map P 0$ is the case:
 * $\ds \prod_{k \mathop = 1}^0 k = 1$

which holds by definition of vacuous product.

Thus $\map P 0$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P r$ is true, where $r \ge 0$, then it logically follows that $\map P {r + 1}$ is true.

So this is the induction hypothesis:
 * $\begin {cases} 1 & : r = 0 \\ r \paren {r - 1}! & : r > 0 \end {cases} \equiv \ds \prod_{k \mathop = 1}^r k$

from which it is to be shown that:
 * $\begin {cases} 1 & : r = 0 \\ \paren {r + 1} r! & : r > 0 \end {cases} \equiv \ds \prod_{k \mathop = 1}^{r + 1} k$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\begin {cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end{cases} \equiv \ds \prod_{k \mathop = 1}^n k$