Ordinal numbers addition

Question

I need to prove that if $\alpha\lt\beta$ are ordinals, then there exists t $\gamma\le\beta$ that $\alpha + \gamma = \beta$

Answer 1

Prove by transfinite induction that
for all b, (a <= b implies exists x with a + x = b).

Base case a = b. a + 0 = b.
Successor case. Show (a <= b implies exists x with a + x = b)
implies (a <= b + 1 implies exists y with a + y = b + 1).
Limit case. Show if b is a limit ordinal then
(for all c < b, (a <= c implies exists x with a + x = c)
implies exists y with a + y = b).

Thus by transfinite induction
for all a,b, (a <= b implies exists y with a + y = b).