HOMEWORK #7, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 3.3.1 on page 87). Show that if K is compact,
then sup K and inf K both exist and are elements of K .
Problem 2 (Exercise 3.3.3 on page 88). Show that the Cantor set
dened in Secti
HW #2 Solutions (Math 323)
4.1 (a) 1, 2, 3; (b) 1, 2, 3; (c) 7, 8, 9; (d) , 4, 5; (e) 1, 2, 3; (f) 0, 1, 2; (g) 3, 4, 5; (h) NBA; (i) 1, 2, 3
note that 1 is the sup of this set since are looking
at the intersection of all the intervals, which is [0, 1];
HW #1 Solutions (Math 323)
1.2) Clearly this holds for n = 1, so we may assume it holds for n and prove that it holds for n + 1:
3 + 11 + + (8n 5) + (8n + 3) = 4n2 n + (8n + 3) = 4(n + 1)2 (n + 1).
1.4) a) For n = 1, 2, 3, 4 the sums are 1, 4, 9, 16 so it
HW #5 Solutions (Math 323)
!
!
!
! (n+1)4 2n
1
4 1
14.1) a) The Ratio Test gives ! sn+1
sn ! = n4 2n+1 = (1 + 1/n) 2 2 < 1 so the series converges.
!
!
!
!
2n+1 n!
2
b) The Ratio Test gives ! sn+1
sn ! = 2n (n+1)! = n+1 0 < 1 so the series converges.
!
!
HOMEWORK 1
MARIUS IONESCU
Problem 1 (Exercise 1.2.1 on page 11).
(1) Prove that 3 is irra
tional. Does a similar argument work to show 6 is irrational?
(2) Where does the proof of Theorem 1.1.1 break down if we try to
use it to prove that 4 is irrational?
HOMEWORK #4, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 2.4.2 on page 54).
(a) Prove that the sequence
dened by x1 = 3 and
1
xn+1 =
4 xn
converges.
(b) Now that we know lim xn exists, explain why lim xn+1 must also
exist and equal the s
HOMEWORK #3, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 2.3.1 on page 49). Show that the constant sequence (a, a, a, a, . . . ) converges to a.
Problem 2 (Exercise 2.3.2 on page 49). Let xn 0 for all n N
(a) If (xn ) 0, show that ( xn )
HOMEWORK # 2
MARIUS IONESCU
Problem 1 (Exercise 1.4.2 on page 27). Recall that I stands for the
set of irrational numbers.
(a) Show that if a, b Q, then ab and a + b are elements of Q as
well.
(b) Show that if a Q and t I, then a + t I and at I as long
as
HOMEWORK #5, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 2.7.1 on page 67; This problem is worth
double the points (12points). Proving the Alternating Series Test
(Theorem 2.7.7) amounts to showing that the sequence of partial sums
s n =
HOMEWORK #6, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 3.2.2 on page 83). Let
B=
(a)
(b)
(c)
(d)
(1)n n
: n = 1, 2, 3, . . .
n+1
.
Find the limit points of B .
Is B a closed set?
Is B an open set?
Find B .
Problem 2 (Exercise 3.2.3 on
HOMEWORK #10, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 6.2.3 on page 161). Consider the sequence of
functions
x
hn (x) =
1 + xn
over the domain [0, ).
(a) Find the pointwise limit of (hn ) on [0, ).
(b) Explain how we know that the co
HOMEWORK #9, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 5.2.2 on page 136).
(a) Use Denition 5.2.1
to product the proper formula for the derivative of f (x) = 1/x.
(b) Combine the result of (a) with the chain rule (Theorem 5.2.5)
to sup
HOMEWORK #7, REAL ANALYSIS I, FALL 2012
MARIUS IONESCU
Problem 1 (Exercise 4.3.5 on page 113). In Theorem 4.3.4, statement
(iv) says that f (x)/g (x) is continuous at c if both f and g are, provided
that the quotient is dened. Show that if g is continuous
HW #6 Solutions (Math 323)
17.1) a) f g, f+ g : x 4; f g = f (g(x) : |x| 2; g f = g(f (x) : x 4.
b) f g(0) = 2, g f (0) =
4, f g(1) = 3, g f (1) = 3, f g(2) = 0, g f (2) = 2.
c) No, thats what part (b) shows.
d) No for
f (g(3), yes for g(f (3).
17.2) a) (