MATH 3150 HOMEWORK 4
Update 9/30: There was a typo in Problem 4.
Problem 1. Prove the converse in Proposition 1.5.5.(i). In other words, Suppose that if xn
is a sequence in R which is bounded below, and a R has the following properties:
(a) For all > 0, t
Math 3150 Midterm Exam Solutions Fall 2013
Problem 1. Complete the following statements.
(a) x is the limit of a sequence xk in a metric space (M, d) if and only if:
Solution. For all > 0 there exists N N such that for all n N , d(xn , x) < .
(b) x is a c
MATH 3150 HOMEWORK 9
Problem 1 (p. 191, #4). Let f : A Rn R be continuous, x, y A and c : [0, 1]
A Rn be a continuous curve joining x and y . Show that along this curve, f attains its
maximum and minimum values (among all values along the curve).
Solutio
MATH 3150 HOMEWORK 10
Problem 1. Prove the following wellknown calculus rule, using the denition of the derivative: If f (x) = xn , n N, then f is dierentiable and f (x) = nxn1 .
Solution. We want to show that
lim
h0
(x + h)n xn
nxn1 = 0.
h
However,
n
n
MATH 3150 HOMEWORK 8
Problem 1 (p. 184, #1). Let f : R R be continuous. Which of the following sets are
necessarily closed, open, compact, or connected?
(a) x R f (x) = 0 .
(b) x R f (x) > 1 .
(c) f (x) R x 0 .
(d) f (x) R 0 x 1 .
Solution.
(a) The set is
MATH 3150 HOMEWORK 1
Problem 1. Prove the following proposition.
Proposition. In an ordered eld, the following properties hold:
(i) Unique identities. If a + x = a for every a, then x = 0. If a x = a for
every a, then x = 1.
(ii) Unique inverses. If a + x
MATH 3150 HOMEWORK 6
Updated 10/28: added Problem 5.
Problem 1 (p. 155 #1). Show that A M is sequentially compact if and only if every
innite subset of A has an accumulation point in A.
Solution. Suppose A is sequentially compact, and let B A be an innite
MATH 3150 FINAL EXAM PRACTICE PROBLEMS FALL 2013
Problem 1.
(a) Give an example of a connected set A Rn such that Rn \ A is not connected.
(b) Give an example of a compact set K Rn which is not connected.
Solution.
(a) A = cfw_x Rn : 1 x 2.
(b) K = cfw_x
MATH 3150 HOMEWORK 2
Problem 1. Dene xn inductively by x1 = 2, xn = 2 + xn1 . This is shown in Example 1.2.10 to be
increasing and bounded. Let = limn xn .
(a) Show that is a root of 2 2 = 0.
(b) Find .
Solution. (a) Since xn is increasing and bounded, it
MATH 3150 HOMEWORK 7
Update 11/2: Typo xed in problem 5.
Update 11/4: Hint added in problem 4.
Problem 1 (p. 172, #1). Which of the following sets are connected? Which are compact?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(x1 , x2 ) R2 x1  1
x Rn
x 10
n
MATH 3150 HOMEWORK 3
Problem 1 (p. 52, #4).
(a) Let xn be a Cauchy sequence. Suppose that for every > 0 there is some n > 1/ such
that xn  < . Prove that xn 0.
(b) Show that the hypothesis that xn be Cauchy is necessary, by coming up with an example
of
MATH 3150 HOMEWORK 5. DUE 10/16
Update 10/9: Added #7, 8 and 9, as well as parts (d)(f) of #3 and part (i) of #6.
Problem 1 (p. 108, #4). Let B Rn be any set. Dene C = x Rn d(x, y ) < 1 for some y B .
Show that C is open.
Solution. Let x C ; we must show
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REAL ANALYSIS
MATH 3150  32107, SEC01, Spring 2016
Instructor: Dr. Brendan McLellan
email: [email protected]
Office: 441 Lake
Office hours: Wednesday 2:005:00
(or by Appointment)
Meeting times and location: TF 9:5011:30 am, 30 Behrakis
Textbook: Elem