Real Analysis Exam 1 Questions with Verified Answers

Real Analysis Exam 1 Questions with
Verified Answers

Infimum - ANSWERS-greatest lower bound
-a real number i is the infimum for a set A ⊆ R if:
1.) i is a lower bound on A (i ≤ a for all a ∈ A)
2.) if l is any lower bound on A, i ≥ l

Set - ANSWERSa collection of objects

Elements - ANSWERSobjects in a set

De Morgan's Law - ANSWERS-(A ∩ B)^c = A^c U B^c
-(A U B)^c = A^c ∩ B^c

Triangle Inequality Theorem - ANSWERS-what it is: |a + b| ≤ |a| + |b|
-used as: |a - b| ≤ |a| + |b|
-also used as: |a - b| = |a - c + c - b| = |(a - c) + (c - b)| ≤ |a - c| + |c - b|

Reverse Triangle Inequality - ANSWERS-what it is: ||a| - |b|| ≤ |a - b|
-used as: Because |a| - |b| ≤ ||a| - |b||, this implies |a| - |b| ≤ |a - b|

Proof by contrapositive - ANSWERSassume not q, show not p

proof by contradiction - ANSWERSAssume p is true and q is false. Show contradiction
exists.

~(P V Q) = - ANSWERS~P ^ ~Q
(not P and not Q)

~(P ^ Q) = - ANSWERS~P V ~Q
(not P or not Q)

Bounded above - ANSWERSA set A ⊆ R is bounded above if there exists b ∈ R such
that a ≤ b for all a ∈ A
** The number b is called an upper bound for A.

Bounded below - ANSWERSA set A ⊆ R is bounded below if there exists l ∈ R such
that l ≤ a for all a ∈ A
**The number l is called a lower bound for A.

Supremum - ANSWERS-least upper bound

, -a real number s is the supremum for a set A ⊆ R if:
1.) s is an upper bound on A (s≥a for all a ∈ A)
2.) if b is any upper bound on A, s ≤ b

Axiom of Completeness - ANSWERSEvery nonempty set of real numbers that is
bounded above has a least upper bound (supremum in R).

Lemma 1.3.8 - ANSWERS-supremums: Assume s ∈ R is an upper bound for a set A ⊆
R. Then, s = supA iff, for every ε > 0, there exists a ∈ A satisfying s - ε < a.
-infimums: Assume i ∈ R is a lower bound on A ⊆ R. Then, i = infA iff, for all ε > 0, there
exists a ∈ A such that i + ε > a.

Theorem 1.4.1 (Nested Interval Property) - ANSWERSFor each n ∈ N, assume we are
given a closed interval In = [an, bn] = {x ∈ R: an ≤ x ≤ bn}. Assume also that each In
contains In+1. Then, the resulting nested sequence of closed intervals
I1 ⊇I2⊇I3 ⊇I4 ⊇···
has a nonempty intersection. In other words,
∞
∩ In ≠∅
n=1
-Note: NIP requires nested intervals to be closed and bounded.

Theorem 1.4.2: Archimedean Property - ANSWERS(i) Given any number x ∈ R, there
exists n ∈ N satisfying n > x (N are not bounded above)
(ii) Given any real number y > 0, there exists an n ∈ N satisfying 1/n < y (can make 1/n
as small as we want)

Theorem 1.4.3: Density of Q in R - ANSWERSFor every two real numbers a and b with
a<b, there exists a rational number r satisfying a < r < b.

Corollary 1.4.4: Density of I in R - ANSWERSFor every two real numbers a and b with a
< b, there exists an irrational number t satisfying a < t < b.

One-to-One - ANSWERSA function f : A → B from A to B is called one-to-one if a1 ≠ a2
in A implies f(a1) ≠ f(a2) in B.

Onto - ANSWERSGiven any b ∈ B, it is possible to find an element a ∈ A for which f(a)
= b.

Cardinality - ANSWERS-A set A has the same cardinality (i. e. size) as B if there exists
f: A->B that is 1-1 and onto
- A ~ B means "A has the same cardinality as B"

Countable - ANSWERS-A set A is countable if N ~ A. An infinite set that is not
countable is called an uncountable set.