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Is every binary representation, i.e. sequence of rational numbers of the form: (91, 92, 93, ...) = (an 2", an 2" +an-1 2-1, an 2" +an-1.2-1+an-2-2-2, ...) (for all i E {n, n - 1, n - 2,..., 2, 1, 0, ...}, ai € {0,1}) necessarily a Cauchy sequence? The answer is: yes! Prove it, and thus give merit to the statement: every binary representation is a real number, and every real number has a binary representation (although not unique!).¹ 10

Is every binary representation, i.e. sequence of rational numbers of the form: (91, 92, 93, ...) = (an 2", an 2" +an-1 2-1, an 2" +an-1.2-1+an-2-2-2, ...) (for all i E {n, n - 1, n - 2,..., 2, 1, 0, ...}, ai € {0,1}) necessarily a Cauchy sequence? The answer is: yes! Prove it, and thus give merit to the statement: every binary representation is a real number, and every real number has a binary representation (although not unique!).¹ 10

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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Every binary representation is actually Cauchy, and thus, a real num- ber. Is every binary representation, i.e. sequence of rational numbers of the form: (91, 92, 93, ...) = (an 2", an 2" +an-1 2-1, an 2n+an-1.2-1+an-2-2-2,...) (for all i E {n, n - 1, n - 2,..., 2, 1, 0, ...}, ai = {0,1}) necessarily a Cauchy sequence? The answer is: yes! Prove it, and thus give merit to the statement: every binary representation is a real number, and every real number has a binary representation (although not unique!).10
Transcribed Image Text:Every binary representation is actually Cauchy, and thus, a real num- ber. Is every binary representation, i.e. sequence of rational numbers of the form: (91, 92, 93, ...) = (an 2", an 2" +an-1 2-1, an 2n+an-1.2-1+an-2-2-2,...) (for all i E {n, n - 1, n - 2,..., 2, 1, 0, ...}, ai = {0,1}) necessarily a Cauchy sequence? The answer is: yes! Prove it, and thus give merit to the statement: every binary representation is a real number, and every real number has a binary representation (although not unique!).10
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