130 pts.
Problem 1. State the following.
(1.) The Well-Ordering Property of N.
(2.) The Principle of Strong Induction.
(3.) The definition of “Denumerable set.”
(4.) Cantor’s Theorem.
(5.) The Triangle Inequality. (The complete version.)
(6.) Let A ⊆ R be a nonempty set. State the definitions of “an upper bound
for A” and sup(A).
(7.) The Completeness Property of R.
(8.) The Archimedean Property.
(9.) The Characterization Theorem for intervals.
(10.) The Nested Intervals Property.
(11.) The definition of the limit of a sequence {xn }∞
n=1 .
(12.) The Squeeze Theorem for sequences.
(13.) The Monotone Convergence Theorem.
1
150 pts.
Problem 2. In each part, decide if the given statement is True or False.
(1.) If S is an infinite set and T $ S, it is possible there is a bijection from T
to S.
(2.) The set Q of rational numbers is countable.
(3.) The set R of real numbers is countable.
(4.) If there is a surjection N → S, then S is denumerable.
(5.) If S ⊆ N, then S is countable.
(6.) If a ≤ b and c ≤ d then a − c ≤ b − d.
(7.) If a ≤ b and c ≤ d then a − d ≤ b − c.
(8.) If a ≤ b and c ∈ R then ca ≤ cb.
∞
(9.) If {xn }∞
n=1 and {yn }n=1 are convergent sequences then
lim xn
xn
= n→∞ .
n→∞ yn
lim yn
lim
n→∞
(10.) A convergent sequence is bounded.
(11.) A bounded sequence is convergent.
(12.) Every nonempty set A ⊆ R has a supremum and an infimum.
(13.) The sequence {(−1)n }∞
n=1 converges.
∞
(14.) Let {xn }∞
n=1 and {yn }n=1 be sequences. If {xn + yn } is convergent, then
{xn } and {yn } are convergent.
(15.) If S is a nonempty set in R that is bounded above, there is a sequence
{sn }∞
n=1 is S that converges to sup(S)
40 pts.
Problem 3. Suppose that a < b. Define y1 = a and y2 = b and let yn be
defined recursively by
yn =
2
1
yn−1 + yn−2 ,
3
3
Show by induction that a ≤ yn ≤ b for all n ∈ N.
2
n ≥ 3.
40 pts.
Problem 4. What is
sup{1 − 1/n | n ∈ N}?
Justify your answer in detail. What major property of the real numbers was
necessary to justify your answer?
40 pts.
Problem 5. Let x1 = 8 and let xn be defined recursively for n ≥ 2 by
xn+1 =
1
xn + 2.
2
Use the Monotone Convergence Theorem to prove that the sequence {xn } converges. Find the limit of the sequence.
3
EXAM
Exam 1
Math 4350-201, Summer II, 2013
July 26, 2013
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 5 problems. There are 400 points
total.
Good luck!