Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #1 Solutions
1. Prove that Q is closed under addition and multiplication. That is, show that if x, y Q, then x + y Q
and xy Q.
Solution: Since x, y Q, we have x = a , y =
b
x+y =
c
d
with a, b, c, d
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #4
Questions from Rudin:
Chapter 2 1,4,9,10,11
1) To see that the empty set is contained in an arbitrary set A, we need to check that every element of the
empty set is in A. Done! The empty set has n
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #3
1. Let z = a + bi be a complex number which for the moment we will think of as a vector in R2 given by
(a, b). Let (z ) denote the angle between the positive x-axis and z satisfying 0 (z ) < 2 . (
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #2 Solutions
1. Recall the notion of a relation R on a set S as dened in class. We further dene the following
concepts:
A relation R is called reective if xRx is true for all x S .
A relation R is
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #5 Solutions
Questions from Rudin:
Chapter 2 12,15
1
Solution: (#12) Let cfw_U be an open cover of K . Thus there is some U containing 0. But since cfw_ n
1
converges to 0 and U is open, all but ni
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #6
Questions from Rudin: Chapter 2: 8, 13
Solution: #8: Yes, every point of an open set U of R2 is a limit point of U . To see this, take x U . Then
since U is open, there exists r > 0 such that Nr (
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #10
1. Let f and g be Riemann integrable functions on [a, b]. If f (x) g (x) for all x [a, b], prove that
b
b
f a g.
a
Solution: Let P = cfw_x0 , . . . , xn be a partition of [a, b]. Since f (x) g (
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #9
Questions from Rudin:
Chapter 5: 1, 12, 22(a,b) try c and d as well! (but not required)
Solution: #1: We will prove that f (x) is identically zero and thus f (x) is constant. To see this, we need
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #8 solutions
Questions from Rudin:
Chapter 4: 2,3,4,18 (replace has a simple discontinuity with is not continuous)
Solution: #2:
Lemma 0.1. Let E X be a subset of a metric space and let z be some ele
Introduction to Analysis MA 511 Fall 2011 R. Pollack
HW #7 Solutions
Questions from Rudin: Chapter 3: 6(a,b), 9, 10, 23, 24(a,b)
Solution: #6(a) We have
n
n
sn =
ai =
i=1
i+1
i=
n+1
1.
i=1
Since sn is unbounded, this sequence must be divergent, and thus
a