Show that

Chapter 2, Problem 35E

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Problem 35E

Show that |R × R| = |R|. [Hint: Use the Schröder- Bernstein theorem to show that |(0, 1) × (0, 1)| = (0, 1)|. To construct an injection from (0, 1) × (0, 1) to (0, 1), suppose that (x, y) ∈ (0, 1) × (0, 1). Map (x, y) to the number with decimal expansion formed by alternating between the digits in the decimal expansions of x and y, which do not end with an infinite string of 9s.]

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