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Define a digit to be an element of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Say that a sequence of digits eventually repeats if there are positive integers N and r such that a+ = a_ for all n > N. Show that a real number in [0, 1] is rational If and only if it has a representation 0. aaa..... For some eventually repeating sequence {a}

Define a digit to be an element of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Say that a sequence of digits eventually repeats if there are positive integers N and r such that a+ = a_ for all n > N. Show that a real number in [0, 1] is rational If and only if it has a representation 0. aaa..... For some eventually repeating sequence {a}

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Define a digit to be an element of the set \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Say that a sequence of digits eventually repeats if there are positive integers \(N\) and \(r\) such that \(a_{n+r} = a_n\) for all \(n > N\). Show that a real number in \([0,1]\) is rational If and only if it has a representation \(0.a_1a_2a_3\ldots\) for some eventually repeating sequence \(\{a_n\}\).
Transcribed Image Text:Define a digit to be an element of the set \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Say that a sequence of digits eventually repeats if there are positive integers \(N\) and \(r\) such that \(a_{n+r} = a_n\) for all \(n > N\). Show that a real number in \([0,1]\) is rational If and only if it has a representation \(0.a_1a_2a_3\ldots\) for some eventually repeating sequence \(\{a_n\}\).
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