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Prove or disprove: a countable set of countable sets is countable.

Solution:

The argument just used for Exercise 1.2 applied to any two countable sets as row and column indices gives the general fact: a countable set of countable sets is countable.

Decimal form of numbers. The numbers between 0 and 1 can each be represented as a decimal point followed by an infinite string of digits, each digit being one of 1, 2, 3, ..., 9, 0.

Rational numbers repeat themselves endlessly after some point: (for example 1/4 is .250* or also .249* where the star means that you repeat the starred digit endlessly) 1/3 is .3*, 1/7 = .(142857)*, 57/100= .570*.

 Irrational numbers do not do so.