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
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 \(\{a_n\}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F7e8525a4-42d9-4579-8e11-11078563c3ae%2Fk09pfl_processed.png&w=3840&q=75)
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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