0.999... - Proofs From The Construction of The Real Numbers - Cauchy Sequences

Cauchy Sequences

Further information: Cauchy sequence

Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |xy|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xmxn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)

If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xnyn) has the limit 0. Truncations of the decimal number b0.b1b2b3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers

left(1 - 0, 1 - {9 over 10}, 1 - {99 over 100}, dotsright)
= left(1, {1 over 10}, {1 over 100}, dots right)

has the limit 0. Considering the nth term of the sequence, for n=0,1,2,..., it must therefore be shown that

This limit is plain; one possible proof is that for ε = a/b > 0 one can take N = b in the definition of the limit of a sequence. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.

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