ECON6003/6703
Assigment 1
Submission Deadline: 5:00pm, 18 March
1. It is a fact that a subset A of RK is closed and bounded⇔ every sequence in A has
a subsequence which converges to a point of A.
Prove the⇒ part of this claim when A ⊂ R.
2. Given the following sequence
0, 1, 0,
1
2
, 0,
1
3
, 0, · · · , 0, 1
n
, · · · ,
does it converge or bounded? Why?
3. In each case, give examples of two functions f and g such that neither f is continu-
ous nor g is continuous but a). f + g is continuous and b). f × g is continuous.
4. For any two vectors a,b ∈ RK if a 6= b, then there exists ε > 0, such that
Bε(a) ∩ Bε(b) = ∅
5. A set is said to be a singleton if it has just one element. Pick any x ∈ RK and consider
the singleton {x}. Prove that {x} is a closed set. Next show that for any given finitely
many points in x0; x1, · · · xk ∈ RK, finite set S = {x0; x1, · · · xk} is a closed set.
(Hint: You can use the various facts on closed and open sets in Lec 3.)