Use a graph to determine where a function is increasing, decreasing, or constant | College Algebra |
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Suppose that the set An is denumerable for each n ∈ N. We may then list the elements of each set: An = an1, 4n2, 4n3, 4n4, .... Now list the elements of the sets A1, A2, A3, as follows: A1 = a11, a12, a13, a14, ..., A2 = a21, a22, a23, a24, ..., A3 = a31, a32, a33, a34, ....Use this construction to prove that ∪ An is a denumerable set: n∈N. This result is often stated as "A countable union of countable sets is countable.
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