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Unformatted text preview: a man.
(b) Case II: The penultimate person is a man. Now consider the ﬁrst
k people in line. The ﬁrst of these is a woman and the last of these
is the penultimate man. According to the induction hypothesis,
somewhere in this line of k people we have a woman directly in front
of a man.
QED 3) Prove:
1− 1
2 1− 1
4 1− 1
8 ··· 1 − 1
2n ≥ 1
1
+
4 2n+1 for all positive integers n.
Proof by induction on n:
1. Base case: Prove true for n = 1. LHS = 1 −
So LHS ≥ RHS. Done with base case. 1
2 1
= 2 . RHS...
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 Fall '08
 HOLMES
 Math, Integers

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