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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
40. Assume that lim
x
ð1Þ
n
1
5

= L, where L is a real number. Then for every ε > 0 there exists an N > 0
such that 1Þ
n
1
5
Lj< ε for all n > N. This implies that ε < L ð1Þ
n
1
5
< ε, i.e.,
ε + ð1Þ
n
1
5
< L < ε + ð1Þ
n
1
5
:
As these inequalities will hold true for a ll n > N, they will hold true for odd an d ev en value s o f n.
Thus one has to consider the following two sets of inequalities:
ε +
1
5
< L < ε +
1
5
and
ε
1
5
< L < ε
1
5
:
As ε can be any positive number, they will have to hold true even when ε = 1/5. In this case the first
set of inequalities will yield 0 < L and the second L < 0. Cle arly th is is impos sible. Therefore the
limit of the sequenc ...
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Publisher Resources

ISBN: 9780123822178