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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_01” 2011/5/22 12:15 page7—#7
Basic Properties of the Real Number System 1-7
Proof Since y x > 0 by the Archimedean property, there exists a
positive integer n such that
n(y x)>1 or that 1 + nx < ny
By Theorem 1.2 .17 (applied to the real number nx), there is an integer
m such that m nx < m + 1. Hence, nx < m + 1 nx + 1 < ny or
that x <(m + 1)/n < y and so r = (m + 1)/n meets the requirement
of the theorem.
The conclusion of the theorem can also be expressed by saying that
the set of all rational numbers
Q is dense in R.
Examples 1.2.16 If both x, y are rationals, then (x + y)/2 is also
rational and clearly lies between x and ...
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Publisher Resources

ISBN: 9781299447561Publisher Website