Real Analysis 501:
Solution to Homework 3
Chapter 3: 8
Example 4. Consider a function f : [0, 1] ! [0, 1]
<0, if x 2 Q,
f (x) = 1
: , if x = in the lowest term.
Determine the continuity of this function.
We claim that f dened above is (1) discon
In the second part of this section, we will discuss numerical series.
3.2.1 Denition and Convergence of a Series
Denition 1. Let cfw_an be a sequence of real or complex numbers. The symbol an is called
an (innite) series. Given a series an
Mathematics is the queen of the sciences by Carl Friedrich Gauss (1865).
Chapter 2 Basic Topology
2.1 From Finite to Countable Sets
Theorem 1. Every innite subset of a countable set is countable.
Theorem 2. A countable union of a sequence of countable set
due Friday, November 16
In the rst two problems the setting is an arbitrary measure space.
1. Let f be a nonnegative measurable function such that X f d = 0.
Prove that f = 0 almost everywhere.
Hint: Consider the sets En = cfw_x | f
due Friday, November 9
1. (Fat Cantor sets)
Let t be any real number satisfying 0 < t < 1/3. Let F0 = [0, 1].
Let F1 be the closed set consisting of two closed intervals obtained by
removing an open interval U1,1 of length t from th
3.1.1 Convergent Sequences
Theorem 1. Every convergent sequence has a unique limit.
Theorem 2. Every convergent sequence is bounded.
Theorem 3. Suppose cfw_xn and cfw_yn are numerical sequences with xn x and yn y. Thus,
1. xn + yn x + y;
Math 501 Midterm, Part I
1. (10 points) Let S denote the set of all sequences of zeros and ones
(Example: .3 = (0,0, 1,0, 1, 1, 1,0, ; . Prove that S is uncountable.
October 8, 2012
we w'nl (alww “bowl” 6%sz C(JMVVl'alle wheel/24, 3 M
. nmwr w
due Friday, November 2
1. Rudin 7.9 (p. 166)
2. Rudin 7.16
3. Show that the sequence cfw_fn in C[0, ] given by fn (x) = sin nx is not
equicontinuous. Do this directly, not by appealing to Arzel`-Ascoli.
4. Let S be a compact subse
due Friday, November 30
1. Rudin 11.3 (p. 332)
2. Rudin 11.11 Hint: This is similar to the proof of Theorem 11.42 but
simpler. Make the appropriate simplications, but be sure to include
enough details to convince me that you know wh
Final Exam December 13, 2012
(a) (3 points) State the simple characterization of Riemann integra-
bility of a function f that involves e and upper and lower sums
and was used by us repeatedly to establish Riemann integrab
due Friday, September 21
1. Let cfw_xn be a sequence of real numbers, and let E be the set of all
limits of subsequences of cfw_xn . E is a subset of the extended real line.
(a) Prove that E is not empty.
(b) Prove that if E = cfw_x
Chapter 4: Continuity
In this section, we will begin to study functions. Our focus is the metric properties of functions.
4.1 Limits of Functions
Suppose f : R R is a function. We say limxx0 f (x) = L if
> 0, = ( , x0 ) > 0, s.t. x for which 0 < |x x0 |