1
Econ 2010 Mathematics for Economists
Homework 4 - Suggested Solutions1
Question 1
Prove that the image of a continuous function on a compact set is compact. Find an example
of a closed, bounded set and continuous function such that the image of that set
1
Econ 2010 Mathematics for Economist
Homework 1 - Suggested Solutions1
Question 1
In the rst lecture we claimed that, if (X, +, , ) is and ordered eld and we dened the
absolute value function | | : X X as
|x| = x if x 0
= x if x < 0
Then the
triangle ine
1
Econ 2010 Mathematics for Economists
Homework 2 - Suggested Solutions1
Question 1
Prove the following properties of closures and interior points
Note: I assume the metric topology on
M
in this solution set. However, the same results
hold under general t
Econ 2010 Mathematics for Economists
1
Homework 3 - Suggested Solutions1
Question 1
In the lecture notes we stated the following: If a metric space M is separable then there exists
a collection of open sets O such that, for any open subset U of M
U = cfw_
Econ 2010 Mathematics for Economists
1
Homework 5 - Suggested Solutions1
Question 1
Show that, for any nite dimensional linear space, any collection L of linearly independent
vectors can be extended to be a basis of that space (i.e. there exists a basis B
Econ 2010 Mathematics for Economists
1
Homework 6 - Suggested Solutions1
Question 1
Part 1
Prove that, if C is convex and C o = cfw_, C o = cl(C )o
Proof. Since C cl(C ), int(C ) int(cl(C ) holds.
Conversely, let x int(cl(C ). Then there exists r > 0 such
Mathematics For Economists
Mark Dean
Midterm
Thursday 21st October
Question 1 Which of the following problems is guaranteed to have a solution? Either prove, or
give a counter-example
1. A rm chooses a level of input R+ to buy in order to maximize prots.
Mathematics For Economists
Mark Dean
Midterm Fall 2011
Wednesday 26th October
Question 1 Let L : R ! R be a linear functional on R. Is it necessarily the case that L maps
open sets to open sets? If so, prove it. If not, provide a counterexample, and a con
Mathematics For Economists
Mark Dean
Midterm
October 22nd 2013
DONT PANIC!
NOTE: Prove all statements, unless told otherwise!
Question 1 Consider the set of functions C (0 1) dened by
n
o
= |C (0 1)]| () = , R and R
That is, is a subset of the set of co
Suggested Solutions to the Midterm Exam
EC2010 TA Section: Yuya Sasaki
Question 1
1. No solution guaranteed from the given information. Let c > 0 and f (t) = (c + 1)t. Then,
this f is concave and continuous, but maxtR+ [f (t) ct], i.e., maxtR+ t, has no s
MATHEMATICS FOR ECONOMISTS
Midterm - 26 th October
Suggested Solution
Question 1.Let L : be a linear functional on . Is it necessarily the case that L maps open
sets to open sets? If so, prove it. If not, provide a counterexample, and a condition on L
suc
1
Econ 2010 Mathematics for Economists
Midterm
1
Question 1.-
Consider the set of functions F C (0, 1) dened by
F = f |C (0, 1)|f (x) = axb , a A R and b B R
That is, F is a subset of the set of continuous functions dened on (0, 1)
Part 1
Let B = R. Is ev