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 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 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 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 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 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 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