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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
positive numbers ε
1
and ε
2
, choose ε
1
= ε
2
=
1
4
ε: Then
ja
n
L
1
j <
1
4
ε for all n > N
ε
1
ja
n
L
2
j <
1
4
ε for all n > N
ε
2
:
So, if n is large enough, all the terms of the sequence will ultimately be at a distance smaller than
1
4
ε
from the two limits. Let n be larger than both N
ε
1
and N
ε
2
. Usually this is written as let n > N =max{N
ε
1
, N
ε
2
}.
In this case, for all n > N
ja
n
L
1
j <
1
4
ε and ja
n
L
2
j <
1
4
ε:
Thus, for all positive numbers ε
jL
1
L
2
j = L
1
a
n
Þ+ ða
n
L
2
Þj jL
1
a
n
j + ja
n
L
2
j <
1
4
ε +
1
4
ε =
1
2
ε < ε:
This proves that |L
1
L
2
| = 0. Since L
1
= L
2
, the sequence has only one limit.
Exercises
6. If all the terms of the sequence fa
n
g
n 1
are i nteger numbers, can the ...
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Publisher Resources

ISBN: 9780123822178