# Concept derivation guides

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Concept guide



## [Real number: proving in the Cauchy completion that 'repeating nines' and the rational 1 are the same real](/en/derivations/derivation-mko-real-number-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-real-number)

In the reals constructed as equivalence classes of Cauchy sequences of rationals (a rational q embedded as the class ι(q) of its constant sequence), let a_n=1−10^{−n} (n≥1), the finite decimals 0.9, 0.99, 0.999 and so on. Prove that (a_n) is a Cauchy sequence equivalent to the constant sequence 1, so [(a_n)]=ι(1); prove that ι is injective, so the real denoted by the finite decimal 0.999=999/1000 is not 1; and show that the finite decimal 0.375=3/8 is rational and is ι(3/8) in ℝ.

Real number

derivation-mko-real-number-01





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