Math 424: Homework 8
Mary Radclie
due 13 Nov 2013
In Rudin: p. 98: problems 1, 2, 3, 6, 8, 20, 21
1. Here we show that the inverse of f : [0, 2) R2 dened by f (t) =
(cos t, sin t) is not continuous, so that inverses of continuous functions on
non-compact
Math 424: Homework 6
Mary Radclie
due 13 Nov 2013
In Rudin: p. 78: problems 6, 7, 14, 15 (for theorems 3.22, 3.23, 3.25(a),
3.33, and 3.34 only)
1. Let cfw_an , cfw_bn be two sequences. We will say that cfw_an decays more
n
quickly than cfw_bn if limn
Math 424: Homework 7
Mary Radclie
due 20 Nov 2013
In Rudin: p. 78: 7, 9, 10, 13
1. Let cfw_an be a sequence of positive terms. Let p = lim sup log an . Show
log n
that if p > 1, then
an converges. What happens if p < 1?
2. Prove the following modication
Math 424: Homework 6 Solutions
Mary Radclie
due 13 Nov 2013
In Rudin:
6. (a) an =
n+1
n
n
Notice that k=1 ak = n + 1 1 = n + 1 1 , so
the series diverges to innity.
(b) an =
n+1 n
n
1
1
Now, an = n+1 n = (n+1+n)n n3/2 . By Theorems
n
3.25 and 3.28, this i
Math 424: Homework 5
Mary Radclie
due 6 Nov 2013
In Rudin: p. 78: problems 2, 5, 20, 21, 22
1. Suppose cfw_pn is a convergent sequence in a metric space X, with limit p.
Show that the set E = cfw_pn cfw_p is compact.
2. For each of the following sets ,
Math 424: Homework 3
Mary Radclie
due 16 Oct 2013
In Rudin: Page 43-45, Problems 7, 9, 10, 14
Also, complete the following.
1. Prove that Q is dense in R.
2. Suppose C is a collection of intervals in R, with I1 I2 = for any I1 , I2 C
with I1 = I2 . Prove
Math 424: Homework 4
Mary Radclie
due 23 Oct 2013
In Rudin: p. 44-45: 18, 19, 22, 23, 27, 28
Also, complete the following. In all of the following problems, X is a metric
space with metric d.
1. Suppose K is a compact set in X. Let > 0. Show that there ex
Math 424: Homework 2
Mary Radclie
due 9 Oct 2013
In Rudin: Page 43, problems 1, 4
Also, complete the following.
1. Suppose A is countable and f : A B is a function. Show that the image
f (A) is at most countable.
2. Suppose A is countable, and B is a set
Math 424: Homework 1
Mary Radclie
due 2 Oct 2013
In Rudin: Page 21-23, problems 2, 3, 6, 8, 17
Also, complete the following.
1. Suppose x, y are real numbers with x < y. Prove that there are innitely
many rational numbers r satisfying x < r < y.
2. Find t
Math 424: Homework 8
Mary Radclie
due 13 Nov 2013
In Rudin:
1. No. Let f (x) = 0 if x = 0 and 1 for x = 0. Then the limit condition is
satised, but f is not continuous.
2. We need only show that if x is a limit point of E , then f (x) f (E ). Let
x be a l
Math 424: Homework 5
Mary Radclie
due 6 Nov 2013
Problems from Rudin:
2. Note,
n2 + n n =
=
=
=
(
n2 + n n)
n2 + n + n
n2 + n + n
n
+n+n
1
1
2+n+1
nn
1
.
1
1+ n +1
n2
1
n
0 by Theorem 3.20, we can conclude that
1
this). Thus we obtain n2 + n n 2 .
As
1+
Math 424: Homework 7
Mary Radclie
due 20 Nov 2013
In Rudin:
9.
n
n3 z n : Note that as n n 1, we also have n3 1, and thus the radius
of convergence is 1 by Theorem 3.39.
2n n
n! z :
Consider
n
2n
n!
1
n! .
=2n
0, and thus, limn
by Theorem 3.39.
n
By Examp
Math 424: Homework 4 Solutions
Mary Radclie
Problems from Rudin:
18. Yes. There are many constructions of such a set. Here is one.
Let 1 , 2 , . . . denote the set of all rational numbers between 0 and 1.
Write each i in its decimal expansion; we will den
Math 424: Homework 3 Solutions
Mary Radclie
Problems from Rudin:
7. (a) We show that Bn = n Ai , which yields the result. Note that it
i=1
is clear that any limit point for a set Ai is a limit point for Bn , so
clearly n Ai Bn .
i=1
Let us suppose that x
Math 424: Homework 1
Mary Radclie
due 2 Oct 2013
Problems from Rudin:
2. Suppose there is a rational number a/b such that (a/b)2 = 12. We may
assume that a and b have no common factor. Then a2 = 12b2 . Therefore,
a2 is divisible by 3. Since 3 is prime, th
Math 424: Homework 2
Mary Radclie
due 9 Oct 2013
Problems from Rudin:
1. Let A be a set. Then A B A for any set B . Moreover, A = , so
A.
4. No, it is uncountable. As Q is countable, if the set of all irrational numbers
were countable, R could be written