I have just begun using Judson's 2018 Abstract Algebra: theory and applications. In the text, there is a Lemma with the following statement and proof:
The Principle of Mathematical Induction implies that 1 is the least positive natural number.
Proof. Let $S=\{n \in \mathbb{N} | n \ge 1\}$. Then $1 \in S$. Assume that $n \in S$. Since $0<1$, it must be the case that $n=n+0<n+1$. Therefore, $1 \le n < n+1$. Consequently, if $n \in S$, then $n+1$ must also be in $S$, and by the Principle of Mathematical Induction, and $S=\mathbb{N}$. QED.
I am having some issues with this proof.
- The last sentence doesn't seem coherent.
- S is defined as the set of natural numbers $\ge1$, but I don't understand how the proof shows that $S = \mathbb{N}$ to arrive at the conclusion that 1 is the least positive natural number.
Prior to this Lemma we are given the definition of natural numbers $\mathbb{N}=\{1,2,3,...\}$, and as propositions the First and Second Principle of Mathematical Induction, as well as the Principle of Well-Ordering.
Can anyone either help me understand why this proof is correct or otherwise help me fill in what may be missing?