I've given this a shot, but I think it's a bit jumbled. I decided to try induction:Show that for any , the following are equivalent:

a). for every Kuratowski-Inductive .

b). is finite.

The first Kuratowski-Inductive subset of is when . This is finite since .

Assume each of the next sets formed by unions with elements are finite.

ie. .

If we add another we get:

.

This is finite since there are n+1 elements in this set.

Is this right?

Also, do I need to prove the converse? My notes tell me that the converse is by definition.