Writing Proofs in Analysis by Jonathan M. Kane

Author:Jonathan M. Kane
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Note that this result can be used to show that the set of rational numbers is countable. Indeed, the rational numbers can be written as the union where R k are the rational numbers that can be written as a fraction with an integer in the numerator and the positive integer k in the denominator. For example, . Thus, the rational numbers is a countable union of countable sets showing that it is countable. The cardinality of a denumerable set is often written using the symbol (read “Aleph knot” or “Aleph null”). The symbol represents the size of the natural numbers and the size of any set that can be placed in one-to-one correspondence with the natural numbers.

A set which is not a countable set is called uncountable. There is a standard argument that shows that the set of real numbers in the interval (0, 1) is not a countable set. The method, known as a diagonalization argument, first assumes that the real numbers between 0 and 1 can all be written down in a sequence . Then one constructs a real number y between 0 and 1 where the kth digit to the right of the decimal point in y is chosen as follows. If the kth digit to the right of the decimal point of x k is 7, then let the kth digit to the right of the decimal point in y be 3. Otherwise, if the kth digit to the right of the decimal point of x k is not 7, then let the kth digit to the right of the decimal point in y be 7. Figure 6.2 illustrates the process of determining y.

Fig. 6.2Determining y using a diagonalization argument

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