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.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"