Homework # 2, Math 413
Due Friday, January 24, 2014
This homework set has ve (5) problems. Some of them are routine while others require
more thought. You are encouraged to work with others and to ask questions of your instructor; however, you must write
Homework # 6 Solutions, Math 413
1. If we toss a fair coin 2n times, there are a total of 22n possible outcomes
(two possibilites on each toss). The total number of outcomes with exactly
n heads is
(2n)!
(2n)!
=
(2n n)!n!
(n!)2
(this is the 2n choose n co
Homework # 3 Solutions, Math 413
1. Find the pointwise limit for each of the sequences of functions dened
below, then decide if the convergence is uniform.
(a) fj (x) = jx(1 x)j
x [0, 1]
It is clear on this one that if x = 0, 1, then fj (x) = 0 for all j,
Homework # 4 Solutions, Math 413
1. The hypotheses on this one are exactly those of Dinis theorem, which you
proved on HW #3. Thus, we know that the sequence fj actually converges
uniformly to f on [a, b]. Once we know the convergence is uniform, then we
Homework # 5 Solutions, Math 413
1. Here we examine the function
2
f (x) =
e1/x
0
if x = 0
if x = 0
It is clear that we may dierentiate f as many times as we like at any
x = 0. To prove that f is innitely dierentiable at 0 as well, the following
observati
Homework # 2 Solutions, Math 413
1. By the single partition characterization of Riemann integrability, we know
that given > 0, there does exist a partition P such that
k
sup f inf f
Ij
Ij
j=1
j < .
Claim: If f is bounded on some interval I, then
sup |f |
Homework # 1 Solutions, Math 413
1) Suppose f is Riemann integrable on [a, b], say with
b
A=
f (x) dx
a
Then by denition there exists > 0 such that for any two partitions P, P
with mesh m(P ), m(P ) < we have |R(f, P )R(f, P )| < 1. Let P = P be
any xed p
Math 412
Real Analysis II
Winter 2014
Instructor: Dr. Dylan Retsek
Oce: 314 Faculty Oces East
email: dretsek@calpoly.edu
Web: www.calpoly.edu/ dretsek
Phone: (805) 756-2072
Course Objective
Our objective this quarter is twofold. First, we seek to complete
Homework # 5, Math 413
Due Friday, February 21, 2014
This homework set has seven (7) problems. Most of them are routine while some require
more thought. You are encouraged to work with others and to ask questions of your instructor; however, you must writ
Homework # 7, Math 413
Due Friday, March 14, 2014
This homework set has eight (8) problems. Most of them are routine while some require
more thought. You are encouraged to work with others and to ask questions of your instructor; however, you must write u
Homework # 7 Solutions, Math 413
1. Bolzano-Weierstrass Theorem in Rn : Suppose |xj | M for all j N.
Then cfw_xj has a convergent subsequence.
Proof: We proceed by induction on n. Let S = cfw_n N : B-W holds in Rn .
We already know B-W in R1 , so 1 S. As