Math 301 Real Analysis, Spring 2013
Homework B #2
Due: Friday, Feb 8, 2013
1. Let F be a eld and a F. Show that (a) = a.
2. Suppose that a, b are elements in an ordered eld such that a 0 and b 0. Prove that
a b if and only if a2 b2 .
3. Suppose that a, b
Spring 2013
Due Friday 10:00am, May 3
Math 301 Real Analysis Final Exam
Instruction: Do all six problems. Each problem is worth 10 points. Fully justify your argument
to get any credit.
1. Let (sn ), (xn ), (tn ) be sequences of real numbers such that
a.
Math 301: Real Analysis
Spring 2013
Instructor Jung-Jin Lee
Oce 401A Clapp Laboratory
Email [email protected]
Oce Hours Mon, Wed 10 12, Fri 3 5.
Text Elementary Analysis: The Theory of Calculus by Kenneth A. Ross
Sections to be Covered Material in Secti
Math 301 Real Analysis, Spring 2013
Homework B #11
Due: Friday, Apr 26, 2013
b
1. Let f : [a, b] R be a nonnegative continuous function such that
f (x) dx = 0. Prove
a
that f (x) = 0 for all x [a, b].
Hint : Suppose for a contradiction that there is x0 su
Math 301 Real Analysis, Spring 2013
Homework B #10
Due: Friday, Apr 19, 2013
Cn
1. Let n N. If C0 + C1 + C2 + + Cn1 + n+1 = 0, where C0 , C1 , . . . , Cn are real constants,
2
3
n
prove that the equation C0 + C1 x + C2 x2 + + Cn xn = 0 has at least one re
Math 301 Real Analysis, Spring 2013
Homework B #9
Due: Friday, Apr 12, 2013
1. Suppose that f : R R is dierentiable at 0 with f (0) = 1. Suppose further that
f (x + y ) = f (x) + f (y ) for all x, y R. Prove that f is dierentiable everywhere and that
f (x
Math 301 Real Analysis, Spring 2013
Homework B #8
Due: Friday, Apr 5, 2013
1. Let f : R R be a function that is continuous at x = 0. Furthermore, suppose that
f (a + b) = f (a) + f (b) for all real numbers a and b. Show that f is continuous everywhere.
2.
Math 301 Real Analysis, Spring 2013
Homework B #7
Due: Friday, Mar 29, 2013
1
1. Let (ak ) be a sequence with lim |ak | k > 1. Prove that lim |ak | = .
k
k
a2
k
2. Suppose that
and
k=1
3. Show that
k=1
b2
k
converge. Prove that
k=1
ak bk also converges.
k
Math 301 Real Analysis, Spring 2013
Homework B #1
Due: Friday, Feb 1, 2013
1. Let S be a subset of N satisfying the following properties:
i) 2 S ,
ii) if n S , then 1, 2, . . . , n 1 S , and
iii) if n S , then 2n S .
Show that S = N.
2. Let a, r be consta
Math 301 Real Analysis, Spring 2013
Homework B #6
Due: Friday, Mar 8, 2013
1. Let r R. Let (sn ) be a bounded sequence with lim sup sn < r. Show that there is N
such that sn < r for all n > N .
2. Let (sn ) and (tn ) be bounded sequences. Prove that
lim s
Math 301 Real Analysis, Spring 2013
Homework B #5
Due: Friday, Mar 1, 2013
1. Compute, with justication,
1
a. lim (2n + 3n + 4n ) n .
n
1
b. lim (n2 10n) n .
n
c. lim sn , where sn =
n
1
,
n
1
,
n+1
n is odd,
n is even.
2. Suppose that lim sn = and lim tn
Spring 2013
Due Tuesday 10:00am, March 12
Math 301 Real Analysis Midterm Exam
No rewrites will be allowed
1. a. (5 points) Let > 0. Show that there exist m, n Z such that 0 < m + n 2 < .
Hint : Consider the limit of ( 2 1)k as k . You may want to use the