March 7, 2017

# Advanced Calculus: An Introduction to Linear Analysis by Leonard F. Richardson

By Leonard F. Richardson

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Extra info for Advanced Calculus: An Introduction to Linear Analysis

Sample text

But if we agree not to allow endless tails of9's, then decimal expansions of real numbers are unique. Moreover, every infinite decimal representation corresponds to a real number. The reason for this fact is as follows. Consider any infinite decimal expression. d1d2d3 ... dn ... , where dn is the nth digit to the right of the decimal point. d1d2 ... dn. It follows that if m and n are both greater than N, then lxn- Xml < 1 10N ---+ 0 as N ---+ oo. Hence the sequence Xn of truncations of the endless decimal expression to n digits is itself a Cauchy sequence.

9 with n 9's then 1 lxn - 11 = -l()n ---+ 0 as n ---+ oo. But if we agree not to allow endless tails of9's, then decimal expansions of real numbers are unique. Moreover, every infinite decimal representation corresponds to a real number. The reason for this fact is as follows. Consider any infinite decimal expression. d1d2d3 ... dn ... , where dn is the nth digit to the right of the decimal point. d1d2 ... dn. It follows that if m and n are both greater than N, then lxn- Xml < 1 10N ---+ 0 as N ---+ oo.

D11d12d13. d21d22d23. d31 d32d33 · · · d3k · · · Now we obtain a contradiction by constructing a number x E [0, 1) that is not in the sequence Xn. We define x by the digits dk in its decimal expansion. If d11 =f=. 0, we let d1 = 0. If du = 0, let d 1 = 1. If d22 =f. 0, we let d2 = 0. If d22 = 0, we let d2 = 1. In general, if dkk =f=. 0, we let dk = 0, but if dkk = 0, then we let dk = 1. d1d2d3 ... dk ... E [0, 1), yet x tj. { Xn} since for all n, x differs from Xn in the nth decimal digit.