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Math 314W
Fall 2013
R. Kantrowitz
Homework #25
1. 2.3.3. Show that if xn yn zn for all n N, and if lim xn = lim zn = l, then lim yn = l.
Proof. For > 0, N1 , N2 N such that |xn l| < n N1 and |zn l| < n N2 . So, for
n maxcfw_N1 , N2 , both are satised, spe
Math 314W
Fall 2013
R. Kantrowitz
Homework #36
1. [4.3.12.(a)]
Prove that the characteristic function g : [0, 1] R of the Cantor set C fails to be
continuous at any point c C.
Proof. The Cantor set C contains no interval. Thus, for c C, and any > 0, the n
Math 314W
R. Kantrowitz
Exercise 1.4.6(b). For any positive number 0 < b, there is a number R such that 2 = b.
Proof. Introduce the set
S = cfw_s R : 0 < s and s2 < b
Consider a number 0 < t that satises t < mincfw_1, b. Then t < 1 which implies t2 < t an
Math 314W
R. Kantrowitz
x
Consider the subset A = cfw_ n : n N of the eld F(R) of formal rational functions.
a. The element 1 F(R) is a lower bound for A since
x
n
1=
xn
n
is positive for all n N.
b. A does not have an inmum in F(R).
Proof I. Suppose to t
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 2:
September 9 - September 13.
ordered elds; PMI.
Homework #4 (Due in my oce by 3:30 p.m. on Wednesday, September 11)
1. For n N, the symbol Zn denotes the set cfw_[0]n , [1]n , [2]n , . . . , [n 1]n
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Weeks 0 & 1:
August 29/30 - September 6.
Section 1.1; ordered elds; PMI
No class and no oce hours on Friday, September 6.
Homework #1 (Due in my oce by 3:30 p.m. on Monday, September 2)
Dividing a number p
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 3:
September 16 - September 20.
Sections 1.3, 1.4: completeness, Arch. property, density,
b.
Homework #7 (Due in my oce by 3:30 p.m. on Wednesday, September 18)
1. If new operations of addition and mu
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 6:
October 7 - October 11.
Sections 2.2, 2.3.
Your attendance = 3 bonus homework points:
Department of Mathematics Colloquium
Completing a Matrix by Maximizing its Determinant
Gilbert Strang
Professor
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 4:
September 23 - September 27.
density,
b, cardinality.
Reminder: Test 1 is Tuesday, October 1.
Homework #10 (Due in my oce by 3:30 p.m. on Wednesday, September 25)
1. Exercise 1.3.8
2. Exercise 1.3.
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 5:
September 30 - October 4.
1.5 [Cantors theorem]; Section 2.2
Test 1: Tuesday, October 1 available starting at 2:00 p.m.
Coverage is all topics covered thus far, including appropriate parts of chap
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 7:
October 14 - October 18.
Sections 2.3, 2.4.
Have a great fall break!
Homework #18 (Due in my oce by 3:30 p.m. on Wednesday, October 16)
1. Exercise 2.3.8(e). Short answer; no proof.
2. Use the -N d
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 8:
October 21 - October 25.
Sections 2.4, 2.5.
Homework #20 (Due in my oce by 3:30 p.m. on Wednesday, October 23)
1. Suppose that (xn ) is a sequence of real numbers, (yn ) is a bounded sequence of no
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 9:
October 28 - November 1.
Sections 2.6, 2.7 (no rearrangements).
Reminder: Test 2 is Tuesday, November 5.
Homework #23 (Due in my oce by 3:30 p.m. on Wednesday, October 30)
1. Exercise 2.5.3(b),(d)
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 10:
November 4 - November 8.
Sections 3.2; 4.1 (Dirichlet), 4.2.
Test 2: Tuesday, November 5 available starting at 2:00 p.m.
Coverage is all topics since Test 1:
1. appropriate parts of sections 2.2,
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 12:
November 18 - November 22.
EVT (4.4); IVT (4.5); Cantor set (3.1).
Homework #31 (Due in my oce by 3:30 p.m. on Wednesday, November 20)
1. Exercise 4.3.11(d) (Just dene a function; no justication.
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 11:
November 11 - November 15.
Sections 4.2, 4.3, 4.4.
Homework #28 (Due in my oce by 3:30 p.m. on Wednesday, November 13)
1. Provide an example or state impossible. No justication.
a. an open subset
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 13:
December 2 - December 6.
7.6 (sets of measure 0); 4.4 (uniform continuity); 7.2.
Test 3: Monday, December 9.
Test will be available at 1:30 p.m.
No texts, no notes, no electronic access during t
Math 314W / Real Analysis
R. Kantrowitz
Fall 2013
Week 14:
December 9 - December 13.
7.2, 7.3, Lebesgues Theorem (7.6)
Test 3: Monday, December 9.
Test will be available at 1:30 p.m.
No texts, no notes, no electronic access during the test.
Homework #37
MATH 314W
FALL 2013
R. KANTROWITZ
PROPERTIES OF FIELDS A SAMPLING
Suppose that (F, +, ) is a eld with:
o additive identity element 2: a: + z = z + :1: = a: for all a: e F,
o multiplicative identity element u: mu = us: = a: for all :1: e F.
Let a,b,:z:
DifferentialEquations
SecondMidtermStudyGuide
Section1:solutionsofeigenvalues
1) Tworealdistincteigenvalues
Y(t)=k1e1tv1+k2e2tv2
a) Both>0:source
b) Both<0:sink
c) 1>0and1<0:saddle
d) if0isoneoftheeigenvaluesalineofequilibriaindirectionofitseigenvector
d.