1.37 THEOREM. Suppose that A and B are sets.

(i) If A ~ Band B is at most countable, then A is at most countable.

(ii) If A ~ B and A is uncountable, then B is uncountable.

(iii) R is uncountable.

PROOF. (i) Since B is at most countable, choose by Lemma 1.36 a function 9 that takes N onto B. We may suppose that A is nonempty, hence fix an ao E A.

Then f(n) := { g(n)

ao g(n) E A g(n) tt A