Review
Second Hour Exam
The exam will cover sequences, continuity, and uniform continuity. This
is the material discussed in class from the last exam up to last
friday.
1. Suppose that
lim an  a
n 
lim bn  b .
and
n 
Prove that
lim an bn  ab .
n 
2. Prove that a Cauchy sequence is a bounded sequence.
3. Let
an 
1 1
1
1
  ...   ...  .
0 ! 1!
2!
n!
Given any  > 0 find an integer N() such that
|an - am| < 
for n, m  N().
4. Explain, in theoretical mathematical terms, why we can claim the
infinite series

1
n
n 1
3
is a "number".
5. Let {an } be a convegent sequence, converging to a. Let
bn 
a1  a2  ....  an
.
n
Prove that {bn} converges to a.
6. State and prove the Bolzano-Weierstrass Theorem.
7. Give two instances of theorems on continuous functions whose proofs
require the use of the Bolzano-Weierstrass Theorem.
8. Let f(x) = ln(1 + x). Given an  > 0 find a  > 0 such that
if |x - 2| <  then |f(x) - f(2)| < .
One possible answer:  = 3(1-e-).
9. Give a direct proof that f(x) = sin(x) is uniformly continuous on
( ,  ) . That is, given an  > 0 find a  > 0 such that
if
|x - y| <  then |sin(x) - sin(y)| < .
10. Let
n
 2
an   1   , n = 1, 2,
 n
Find the least upper bound for the sequence {an}. Be sure to explain why
the number you claim to be the least upper bound is, in fact, the least
upper bound.
11. Let A and B be two non-empty, bounded sets. Prove that
sup(AB) = max{sup(A), sup(B)}.
12. What would be a "reasonable" definition for sup(A) if A were the
empty set?