MAT321-Lecture Notes
A. TUFA
Exercise 1.1
1. Determine whether the following sets are bounded from below, above, or
both. If so, determine their infimum and/or supremum and find out whether
these infima/suprema are actually minima/maxima.
a) {x ∈ R : x2 + x + 3 > 0}.
b) {2 − (−1)n : n ∈ N}.
2. Find sup A and inf A if any and prove that your answer is correct.
a) A = {2, 3, 16, 19.007, 23}.
1 1
1
b) A = 1, , , · · · =
:n∈N .
2 3
n
c) A = {x ∈ R : x2 < 7}.
d) A = {x ∈ Q : 0 ≤ x ≤
√
2}.
3. Let A = (−1, 2]. Prove that inf A = −1, but min A doesn’t exist.
4. If a ∈ R and a < 0, then prove that sup(aA) = a inf A, where
aA = {ax : x ∈ A}.
5. Let A and B nonempty subsets of R which are both bounded from above.
Define the set A + B as
A + B = {a + b : a ∈ A, b ∈ B}.
Show that sup(A + B) = sup A + sup B.
6. Let {an } be a sequence defined by the following relations.
√
√
a1 = 2, an+1 = 2 + an , ∀n > 1.
Use the principle of mathematical induction to show that
a) an ≤ 2 for all n.
a) an ≤ an+1 for all n.
7. Prove that Q∗ is dense in R.
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