关键词 > ECON6003/6703
ECON6003/6703 Mathematical Methods for Economics Problem Set 1
发布时间:2022-03-11
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Problem Set 1
ECON6003/6703 Mathematical Methods for Economics
Question 1. Starting from page 20 of your Lecture 1 slides, a number of environ- ments are described. In each of the questions, you are asked to identify certain sets. State for each case, whether those sets are open, closed or compact.
Question 2. Given a pair of closed sets A ; B Rm which of the following sets are closed? In each case, give a proof if you claim the set is closed, and a counter example otherwise.
1 A \ B 2 A n B 3 A
B
Question 3. Repeat the above question where the closedness
is replaced with
compactness
.
Question 4. A set is said to be a singleton if it has just one element. Pick any x 2 Rm and consider the singleton fxg. Prove that fxg is a closed set. Next show that for any given finitely many points in x0; x1; : : : ; xk2Rm, finite set S = fx1 ; : : : ; xkg is a closed set. Hint: You can use the various theorems on closed and open sets in Lec 2.
Question 5. Using Theorem 2 Lec 2 notes (Theorem 12.8 from text) and the general results for set operations on page 19 of Lec 1 notes, give a complete proof of Theorem 3, Lec 2 notes.
Question 6. (Tricky) Let A Rm be a compact set with at least two elements. Show that there exists a number s
such that for every " > 0, there exists x; y 2 A such that s
d(x; y) > s
¡ ".