Proof: By Euclid
(related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
- For let any even multitude whatsoever of odd numbers, $AB$, $BC$, $CD$, $DE$, lie together.
- I say that the whole, $AE$, is even.
- For since everyone of $AB$, $BC$, $CD$, $DE$ is odd then, a unit being subtracted from each, everyone of the remainders will be (made) even [Def. 7.7] .
- And hence the sum of them will be even [Prop. 9.21].
- And the multitude of the units is even.
- Thus, the whole $AE$ is also even [Prop. 9.21].
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"