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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
b. We assume that the inequality is true for an arbitrary number n. Thus
1
n + 1
+
1
ðn + 1Þ+ 1
+
::::
+
1
2n
>
1
2
:
c. We need to prove that the inequality holds true for n + 1. We will add fractions with
denominators between (n + 1) + 1 and 2(n + 1). So, we want to prove that
1
ðn + 1Þ+ 1
+
1
ðn + 1Þ+ 2
+
::::
+
1
2n + 1
+
1
2ðn + 1Þ
>
1
2
:
One thing to notice is that the smallest denominator of the fractions in the inductive hypothesis is
n + 1, while the smallest denominator in this step is n + 2. Thus, to make the inequality in the
inductive hypothesis and the left-hand side of the inequality to b e proved start with fractions
having the same denominator, we could rewrite ...
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Publisher Resources

ISBN: 9780123822178