(related to Proposition: Every Integer Is Either Even or Odd)
- Let $m$ be an integer.
- According to the Euclidean lemma about the quotient and remainder there are unique integers $q$ and $r$ such that $m=2q + r$ with $0\le r < 2.$
- Case $r=0$: Then $m=2q$, and $m$ is even by definition.
- Case $r=1$: Then $m=2q+1$, and $m$ is odd by definition.
- Since $r$ must by either $0$ or $1$ because of $0\le r < 2,$ it follows that every integer is either even or odd.
- Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016