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**novice** (The Principle of Mathematical Induction) For each positive integer $\displaystyle n$, let $\displaystyle P(n)$ be a statement. If

(1) $\displaystyle P(1)$ is true and

(2) the implication If $\displaystyle P(k)$, then $\displaystyle P(k+1)$

is true for ever positive integer $\displaystyle k$, then $\displaystyle P(n)$ is true for every positive integer $\displaystyle n$.

*Proof:*Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers $\displaystyle n$ for which $\displaystyle P(n)$ is a false statement. Let

$\displaystyle S = \{n \in \mathbb{N}: P(n)$ is false $\displaystyle \} $

Since $\displaystyle S$ is a nonempty subset of $\displaystyle \mathbb{N}$ , it follows by the Well-Ordering Principle that $\displaystyle S$ contains $\displaystyle s$. Since $\displaystyle P(1)$ is true, $\displaystyle 1 \in S$.

Here I am lost:

Thus $\displaystyle s \geq 2$ and $\displaystyle s-1 \in \mathbb{N}$. Therefore, $\displaystyle s-1 \in S$ and so $\displaystyle P(s-1)$ is a true statement. By condition (2), $\displaystyle P(s)$ is also true

and so $\displaystyle s \not \in S$. This however, contradicts our assumption that $\displaystyle s \in S$.

**Question: What is $\displaystyle s-1$?**