HOMEWORK 1SOLUTIONS1. Page 10(2) For each of the following statements, determine whether it is true orfalse, and justify your answer.(a) Every nonempty set of real numbers that is bounded above hasa largest member.False. As a counter example, consider the interval (1, 2), thisis bounded above by 2, but has no largest member.(b) If S is a nonempty set of postive real numbers, then 0 ≤ inf(S).True
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HOMEWORK 1
SOLUTIONS
1. Page 10
(2) For each of the following statements, determine whether it is true or
false, and justify your answer.
(a) Every nonempty set of real numbers that is bounded above has
a largest member.
False. As a counter example, consider the interval (1, 2), this
is bounded above by 2, but has no largest member.
(b) If S is a nonempty set of postive real numbers, then 0 ≤ inf(S).
True.
Proof. S a nonempty set of positive real numbers means that
∀x ∈ S, x > 0, so 0 is a lower bound for S. This 0 ≤ inf(S),
since by definition, inf(S) is the greatest lower bound of S.
(c) If S is a set of real numbers that is bounded above and B is a
nonempty subset of S, then sup(B) ≤ sup(S).
True.
Proof. B is bounded above by any upper bound of S, so in
particular, sup(S) is an upper bound for B, therefore sup(B) ≤
sup(S) since by definition sup(B) is the least upper bound for
B.
(3) Use the principle of mathematical induction to prove for any n ∈ N,
(1)
nXj
=1
j2 = n(n + 1)(2n + 1)
6
.
Proof. First let us check n = 1, the left hand side of (1) is simply
equal to 1, and the right hand side is equal to (1)(2)(3)
6
= 1, so (1)
holds for n = 1
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