gistfile1.txt

Lemma 2: ("Real Induction") Let A be a subset of real numbers.
If
 (0) α ∈ A
 (1) ∀x ∈ A. ∃y. [x,y) ⊆ A
 (2) ∀xy. [x, y) ⊆ A ⇒ y ∈ A
then [α,∞) ⊆ A.

Proof:
Suppose towards a contradiction that [α,∞) ∖ A is nonempty. Set β = inf ([α,∞) ∖ A).
(1) means we can't have β ∈ A, because it would collide with the infimum.
So β > α and β ∉ A. But then [α,β) ⊆ A, and so β ∉ A, a contradiction.
∎