Economics 519 Exam
University of Arizona Fall 2012
Closed-book part: Don't consult books or notes until you've handed in your solutions to Problems #1, #2, and #3. After you do consult books or notes, your solutions to Problems #1, #2, and #3 will not be
Economics 519: 2012 Exercise Set #4
1. Verify that if the sequence cfw_xn converges and R, then the sequence cfw_xn also converges.
2. Let F : [0, 1] R be the identity function F (x) = x, and for each n N let Fn : [0, 1] R be
the function
Fn (x) =
n+1
x
Economics 519 Exam
University of Arizona
Fall 2011
Closed-book part: Dont consult books or notes until youve handed in your solutions to Problems #1, #2, and #3. After you do consult books or notes, your solutions to Problems #1, #2,
and #3 will not be ac
Economics 519 Exam
University of Arizona
Fall 2010
Closed-book part: Dont consult books or notes until youve handed in your solutions to Problems #1 and #2 . After you do consult book or notes, your solutions to Problems #1 and #2 will
not be accepted.
1.
Economics 519 Exam
University of Arizona
Fall 2009
Closed-book part: Dont consult books or notes until youve handed in your solutions to Problems #1 and #2 . After you do consult book or notes, your solutions to Problems #1 and #2 will
not be accepted.
1.
Economics 519: Exercise Set #3
1. The set R is a vector space under the usual (component-wise) denitions of vector addition
and scalar multiplication:
cfw_x1 , x2 , . . . + cfw_y1 , y2 , . . . := cfw_x1 + y1 , x2 + y2 , . . .
Verify that the set
and
cfw_x
Economics 519: Exercise Set #2
1.
For each of the following norms
on R2 , draw the set cfw_x R2 | x = 1:
x2 + x2
2
1
(a)
x=
(b)
x = |x1 | + |x2 |
(c)
x = maxcfw_|x1 |, |x2 |
2.
Show that each of the following is a vector space:
(a) The set X of all linear
Economics 519: Exercise Set #1
1.
Prove that for a preorder
on a set X , the associated relation
associated relation is an equivalence relation, where
x
2.
x [x
Dene the relation
x&x
on R2 by x
x]
and
x [ x1
is a strict preorder and the
and are dened by
x
Economics 519: 2012 Exercise Set #5
1. Apply the denition of Cauchy sequence directly to prove that the sequences cfw_xn and cfw_yn
dened by xn =
1
n
and yn = 1
1
n
are both Cauchy sequences.
2. Making use of the fact that R is a complete metric space,
Economics 519: 2012 Exercise Set #6
1. Prove that every compact metric space is bounded.
2. Prove that every compact metric space is complete.
3. For each of the following conjectures, provide a counterexample to show that the conjecture is
false:
(a) If
Demand Theory
(The Theory of Consumer Behavior, or Consumer Choice)
This quick summary of demand theory is a preview of the first part of Econ 501A, but it also serves as a prototype or template for other models of decision-making we'll develop in Econ 50
Economics 519 Exam
University of Arizona Fall 2011
Closed-book part: Don't consult books or notes until you've handed in your solutions to Problems #1, #2, and #3. After you do consult books or notes, your solutions to Problems #1, #2, and #3 will not be
Economics 519 Exam
University of Arizona Fall 2010
Closed-book part: Don't consult books or notes until you've handed in your solutions to Problems #1 and #2 . After you do consult book or notes, your solutions to Problems #1 and #2 will not be accepted.
Economics 519 Exam
University of Arizona Fall 2009
Closed-book part: Don't consult books or notes until you've handed in your solutions to Problems #1 and #2 . After you do consult book or notes, your solutions to Problems #1 and #2 will not be accepted.
Economics 519: 2012 Exercise Set #4
1. Verify that if the sequence cfw_xn converges and R, then the sequence cfw_xn also converges. 2. Let F : [0, 1] R be the identity function F (x) = x, and for each n N let Fn : [0, 1] R be the function Fn (x) =
n+1 x
Economics 519: Exercise Set #3
1. The set R is a vector space under the usual (component-wise) definitions of vector addition and scalar multiplication: cfw_x1 , x2 , . . . + cfw_y1 , y2 , . . . := cfw_x1 + y1 , x2 + y2 , . . . Verify that the set
and
cfw
Economics 519: Exercise Set #2
1. (a) (b) (c) For each of the following norms x = x2 + x2 2 1 on R2 , draw the set cfw_x R2 | x = 1:
x = |x1 | + |x2 | x = maxcfw_|x1 |, |x2 |
2.
Show that each of the following is a vector space:
(a) The set X of all linea
Economics 519: Exercise Set #1
1. Prove that for a preorder on a set X, the associated relation is a strict preorder and the
associated relation is an equivalence relation, where x x [x x&x x] and
and are defined by x&x x] .
x x [x
2.
Define the relation
Open and Closed Sets
Denition: A subset S of a metric space (X, d) is open if it contains an open ball about each of
its points i.e., if
x S : > 0 : B (x, ) S.
(1)
Theorem:
(a) and X are open sets.
(b) If S1 , S2 , . . . , Sn are open sets, then n=1 Si is
Norms and Metrics,
Normed Vector Spaces and Metric Spaces
Were going to develop generalizations of the ideas of length (or magnitude) and distance. Well
generalize from Euclidean spaces to more general spaces, such as spaces of functions. We begin
with th
Economics 519: 2010 Exercise Set #6
1. Prove that every compact metric space is bounded.
2. Prove that every compact metric space is complete.
3. Let C denote the unit circle in R2 . In polar coordinates, the points in C are denoted (r, ),
where r = 1 and
Economics 519: 2010 Exercise Set #5
1. Making use of the fact that R is a complete metric space, prove that the unit interval [0, 1] in
R is a complete metric space.
2. Prove that a subset of a complete metric space is itself complete if and only if it is