Real Analysis Test – Chapters 3 & 4
Name:
1. Using an    proof, show that lim  x 2  2 x  3  3 .
x 2
2. Use the definition of uniform continuity to prove that f ( x)  x3 is uniformly
continuous on [2,1] .
3. Determine whether each of the following statements is true or false. If false give a
counter-example, if true, explain briefly:
a. If f :[a, b] 
b. f f :

is Lipschitz then it must be uniformly continuous on [a,b].
is continuous and there exists a sequence  xn  with lim( xn )  
n 
such that lim( f ( xn ))  , then lim( f ( x))  .
n 
x 
c. A function that is uniformly continuous on every bounded interval, is
uniformly continuous on .
d. If f and g are uniformly continuous on D and g ( x)  0 on D, then
uniformly continuous on D.
e. There is a continuous function that
f
is
g
i. maps an open interval to a closed interval.
ii. maps a closed interval to an open interval.
4.
Prove: If f :[0,1]  is monotone and satisfies the intermediate value property,
then f is continuous.
5. Let A 
. We define the derived set of A, denoted A , A   x x is a limit point of A .
A perfect set is a set which is equal to its derived set.
Prove: Every closed set A 
countable.
can be written as a union, A  P
C with P perfect and C
5.
Real Analysis Take-Home Exam
Instructions: SHOW ALL WORK !! Exam is due at 4:15 pm on Tuesday 11/24. Consult no
other person regarding these questions. You are not allowed to use the internet. (You may
use other references including your notes, calculator, Mathematica, and text.) Read the
following paragraph and sign it.
I have neither given nor received assistance on any of these questions. I understand that the
penalty for cheating on this exam is a D in the course.
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7. Let x  [0,1] with ternary expansion  an  . Let N   if none of the an are 1, and
1
otherwise let N  Min{n : an  1} . Let bn  an , for n  N and bN  1 .
2
N
bn
a. Show that  n is independent of the ternary expansion of x (if x has two
n 1 2
expansions).
N
b
b. Show that the function defined by setting f ( x)   nn is continuous and
n 1 2
monotone on [0,1].
c. Show that f is constant on each of the intervals contained in the
complement of the Cantor set and that f maps the Cantor set onto [0,1].
1. Define the following::
a. lim  f ( x)   L where L, a are real numbers.
x a
b. The function f : D 
is continuous at a.
2. Give an example of a function defined for all real numbers such that the set of all
points of discontinuity are:
a. open but not closed;
b. closed but not open;
c. neither open nor closed.
3. Prove: If f :

is continuous and O 
is open, then f 1 (O) is open.
4. Determine whether the following statement is true or false. If true prove, if false give a
counter-example:
If f :[0,1]  is monotone and satisfies the intermediate value property, then f
is continuous on [0,1].
1. Complete the definition:
i. We say that Lim  an   L if and only if
n
ii. Use the above definition to prove that the sequence an 
3n2  3
converges to 3.
n2  2
2. Complete the following definitions:
i. A set A  is open if and only if
ii.
A number a
iii. A set A 
is said to be an accumulation point of A 
if and only if
is compact if and only if
iv. The sequence {an } diverges to   if and only if
v.
We say that the sequence {an } is eventually increasing if and only if
3. True or False: Give a short explanation or a counter-example.
a. If the sequence {an} is bounded, then it converges.
.
b. If S 
is a set with sup S  s 
, then s is an accumulation point of S.
c. Every bounded sequence has a monotone subsequence.
d. If {an} is convergent, then it is contractive.
e. There exists a non-empty, countably infinite, open set.
3. Prove that if a set A 
is open then its complement is closed.
4. Prove that if an  and bn  are Cauchy, then an  bn  is Cauchy using the definition
of a Cauchy sequence.
5. Let an  be a bounded sequence and T :

be any one-to-one function. Prove that if
an  converges to A, then aT ( n ) n1 also converges to A.

6. Prove that if for some k  , a0  a1  a2 
 ak  0 , then

lim a0 n  a1 n  1  a2 n  2 
n 

 ak n  k  0
#1. True or False: If false, give a counter-example/explain.
1. Every infinite sequence of real numbers has a convergent subsequence.
2. If {an} is subsequence of {bn} and {bn} is subsequence of {an}, then the two
sequences are equal.

3. If En is closed for each n , then
En is closed.
n 1
4. If {an} and {bn} are sequences such that both {an}and { an bn} diverge, then {bn}
diverges.
5. The sequence {an} converges iff  an  converges to zero.
6. The set {0,2} has exactly two accumulation pts.
7. There is an infinite set which has an infinite number of accumulation points such that
the set and the set of accumulation points are disjoint.
8. There is a countable set which has an uncountable number of accumulation points
such that the set and the set of accumulation points are disjoint.
2. Define each term:
5. Let S be a subset of the real numbers. Then S is closed if and only if
6. Liminf  an 
7. The sequence {an} is subsequence of {bn} if and only if
8. A Cauchy Sequence
9. Let A 
An open cover of A is
 2n 2  n  2 
#3. Prove (Using the  -definition): Lim 
2
2
n 
 n n 
#4. Prove: If Ei is open for each i 

Ei is open.
, then
i 1
#5. Consider the set S of all sequences {an} such that {an} satisfies:
i.
For each n  , an  1 or an  2;
ii.
If an  2, then an1  1.
Determine whether S is countable or uncountable, and then prove your claim.
#6. Prove that every Cauchy sequence is bounded using the definition of a Cauchy
sequence.
Name:
#1. Prove that for each n  ,
n
 (2k  1)  n
k 1
2
.
#2 Determine whether each of the following statements is true or false. If true, prove it. If
false, give a counter-example.
a. For sets A, B  , P  A  B   P  A  P  B 
b. If A,B, and C are sets, then A   B C   ( A  B) ( A  C )
#2 (continued) Determine whether each of the following statements is true or false. If
true, prove it. If false, give a counter-example.
c. Every collection of non-overlapping open intervals is at most countable. [For
example, the collection A  (n, n  1) : n   is non-overlapping since the
intersection of any two intervals in A is empty, whereas A  (n  1, n  1) : n 
not non-overlapping].
#3 Prove that
3 is irrational.
 is
#4. Determine whether
10
is in the Cantor Middle-Thirds set.
13
#5. Given n integers, n , prove that either one of them is a multiple of n or some subcollection of them sum to a multiple of n.
#6. Consider any “map” formed by drawing straight lines in a plane to represent
boundaries. The figure below shows a map with ten countries labeled A through J,
formed using four lines in the plane. Prove that every map formed by drawing n straight
lines n , can be colored with exactly two colors in such a way that adjoining
countries (those with a line segment as a common border) are colored with different
colors.
A
B
H
C
I
J
G
D
E
F