Proof: By Euclid
(related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
- For let any multitude whatsoever of even numbers, $AB$, $BC$, $CD$, $DE$, lie together.
- I say that the whole, $AE$, is even.
- For since everyone of $AB$, $BC$, $CD$, $DE$ is even, it has a half part [Def. 7.6] .
- And hence the whole $AE$ has a half part.
- And an even number is one (which can be) divided in half [Def. 7.6] .
- Thus, $AE$ is even.
- (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"