Let a be a set of real numbers. If a belongs to a, then 1 / 1-A belongs to a and 1 does not belong to A. It is proved that if a belongs to a, then 1-1 / a belongs to a

Let a be a set of real numbers. If a belongs to a, then 1 / 1-A belongs to a and 1 does not belong to A. It is proved that if a belongs to a, then 1-1 / a belongs to a

Because if a belongs to a, then it belongs to a,
By substituting a with 1 / 1-A, 1 / [1 - (1 / (1-A))] belongs to a, and 1-1 / a belongs to a