# Question on induction-1 is the least positive integer

### 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 $$1\leq n, so $$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".

• At this level you need to be very careful to define all of your terms and operations. In particular, how exactly are you defining addition, and how are you defining $\le$? Commented Jul 9, 2013 at 1:01
• I agree with dfeuer - for that matter, how are you defining "integer"? How are you defining "positive"? Commented Jul 9, 2013 at 2:10
• Then the question, as stated, just can't be answered. Please delete your two last comments—they are redundant. Commented Jul 9, 2013 at 2:43
• @dfeuer can you understand the proof? Is it right in your sensation? Commented Jul 9, 2013 at 3:11
• No, HyperGroups. Without the necessary definitions I have no idea what any of it is supposed to mean. Commented Jul 9, 2013 at 3:25

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.