Consider the set of all possible sequences {a_n}, where the terms of each sequence a_1, a_2, a_3 ... are each non-negative integers. What is the cardinality of this set? Prove it.
Hint: If the possible values of a_1, a_2, etc. were restricted to {0,1, 2, 3, 4, 5, 6, 7, 8, 9}, it would be obvious that each sequence could be mapped to exactly one real number in [0, 1], so the cardinality of the set of sequences is AT LEAST as big as C (the cardinality of the real numbers in [0, 1]). But each of the members of the sequence can take on any positive integer value, so it could be larger than C.
This has implications. This cardinality is the same as that of the set of all functions that map rational numbers into rational numbers.
2007-11-08
03:14:09
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