Question on induction-1 is the least positive integer

Question

Question on induction

prove: 1 is the least positive integer.

proof:

Let $A=\left\{x\geq 1\left|x\in Z^+\right.\right\}$, and then $1\in A$, if positive integer $n\in A$, then $n\geq 1$,

Since $1\leq n<n+1$, so $n+1\in A$. By induction, we have $A=Z^+$. So for all positive integer that it is greater than or equal to $1$, and $1$ is the least positive integer.

question:

There is something different from the induction pattern

$\text{when } n=1,\text{...}n=k,\text{...}\text{and } n=k+1,\text{...}$

So, can you explain me a bit more about the proof and the corresponding of the pattern I listed?

I saw this as one first example in the first section of first chapter of one text of mathematical analysis. So I do not know the definition of "integer".

Answer 1

Here's a slight rewording of the given proof in the induction format that you're used to:

Claim: For all $n \in \Bbb{Z}^+,~$ $n\ge 1$.

Proof: We proceed by induction on $n \in \Bbb{Z}^+$.

Base Case: For $n=1$, we have $1 \ge 1$, which works.

Induction Hypothesis: Assume that the claim is true for $n=k$, where $k$ is a positive integer. That is, assume that $k\ge 1$.

It remains to prove the claim true for $n=k+1$. Since $k \in \Bbb{Z}^+$, we know by the definition of the positive integers that $k+1\in \Bbb{Z}^+$. Recall that $k+1>k$. But by the induction hypothesis, we know that $k\ge1$. Hence, $k+1\ge1$, so the claim is true for $n=k+1$. This completes the induction.