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Problem 3 Countable and Uncountable Sets (a) Show that there are uncountably infinite many real numbers...

Problem 3 Countable and Uncountable Sets

(a) Show that there are uncountably infinite many real numbers in the interval (0, 1). (Hint: Prove this by contradiction. Specifically, (i) assume that there are countably infinite real numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii) express each real number x1 between 0 and 1 in decimal expansion; (iii) construct a number y whose digits are either 1 or 2. Can you find a way to choose 1 or 2 such that y is different from all the xis?)

(b) Is the set of all irrational numbers in (0, 1) countably infinite? Please explain your answer.

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