Originally Posted by

**superevilcube** I did a proof, it sounds right to me, but I want to make sure.

**Prove***:* 1 is the smallest element in Z+ (the positive integers)

Suppose not, then there exists some integer $\displaystyle a\in Z+$ such that $\displaystyle a<1$

Let $\displaystyle S=[x\in Z+|x<1]$. By our assumption that $\displaystyle a<1$, $\displaystyle a\in S$. Thus S is not the empty set and by the Well Ordering Principle there exists a smallest element, call it $\displaystyle b$. Since $\displaystyle b<1$ it follows that $\displaystyle b-1<0$.

Thus, $\displaystyle b(b-1)<0$ which implies that $\displaystyle b^2<b$. It follows that $\displaystyle b^2<1$. Thus $\displaystyle b^2\in S$. But $\displaystyle b^2<b$, a contradiction since $\displaystyle b$ is the smallest element in S. Thus 1 must be the smallest element in the positive integers.

Is this right?