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Unformatted text preview: Bo˘azi¸i University
gc
Department of Economics
Fall 2011
EC 521 MATHEMATICAL METHODS FOR ECONOMICS
Problem Set 2
Due 28.10.2010
1. Let (X, d) be a metric space. Deﬁne d1 (x, y ) = d(x,y )
1+d(x,y ) for any x, y ∈ X. (a) Show that d1 is a metric on X.
(b) Show that if S ⊆ X is open in (X, d), then S is open in (X, d1 ).
2. Let (X, d) be a metric space. Let f : R+ → R be a concave and strictly increasing
function with f (0) = 0. Show that (X, f ◦ d) is a metric space.
3. Let (X, d) be a metric space. Let (xn ) and (yn ) be two sequences in X with lim xn = x
and lim yn = y. Show that lim d(xn , yn ) = d(x, y ).
4. Determine whether the following sets are open, closed or neither.
(a) {(1/n, 1/n2 ) : n ∈ N} ∪ {(0, 0)} ⊆ R2
(b) {(x, y, x2 y 2 ) : x2 + y 2 < 1} ⊆ R3
∞ (c) [−n, (n − 1)/n] ⊆ R
n=1
∞ (d) (0, 1/n] ⊆ R
n=1 5. Let (X, d) be a metric space. Let A, B ⊆ X.
(a) Show that Int(A) ∩ Int(B ) = Int(A ∩ B )
(b) Show that Cl(A) ∪ Cl(B ) = Cl(A ∪ B )
6. Let (X, d) be a metric space and let S ⊆ X. Show that x ∈ Bd(S ) if and only if
there exist (xn ) in S and (xn ) in X \S such that lim xn = x = lim xn . 1 7. Let (X, d) be a metric space where d is the discrete metric.
(a) Show that any S ⊆ X is open in (X, d).
(b) Show that any function f : X → Y is continuous.
8. Let (X, dX ) and (Y, dY )be two metric spaces, and let f : X → Y be a continuous
function. Show that f (Cl(A)) ⊆ Cl(f (A)) for any A ⊆ X.
9. Show that f : (0, ∞) → R with f (x) = 1/x is not uniformly continuous.
10. Let (X, dX ) and (Y, dY )be two metric spaces. Then, a function f : X → Y is called
Lipschitz continuous if there exists a K > 0 such that dY (f (x), f (y )) ≤ KdX (x, y )
for any x, y ∈ X . Show that if a function f : X → Y is Lipschitz continuous, then
it is also uniformly continuous. 2 ...
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This note was uploaded on 10/25/2011 for the course ECON 501 taught by Professor Zobuz during the Spring '11 term at Istanbul Technical University.
 Spring '11
 zobuz
 Economics

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