Mathematical Analysis I, 2004a

§9. Some Theorems on Countable Sets 21
Note 4. Observe that the numbers a nn used in the proof of Theorem 3 form
the diagonal of the infinitely extending square composed of all a nm . Therefore,
the method used above is called the diagonal process (due to G. Cantor).
Problems on Countable and Uncountable Sets
1. Prove that if A is countable but B is not, then B − A is uncountable.
[Hint: If B − A were countable, so would be
Use Corollary 1.]
(B − A) ∪ A ⊇ B. (Why?)
2. Let f be a mapping, and A ⊆ D f .Provethat
(i) if A is countable, so is f[A];
(ii) if f is one to one and A is uncountable, so is f[A].
[Hints: (i) If A = {u n },then
f[A] ={f(u 1 ),f(u 2 ), ..., f(u n ),...}.
(ii) If f[A] were countable, so would be f −1 [f[A]], by (i). Verify that
here; cf. Problem 7 in §§4–7.]
f −1 [f[A]] = A
3. Let a, b be real numbers (a