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ANALYSIS AND OPTIMIZATION, PROBLEM SET 5
Due Friday 3/13, 5pm
Problem 1. Let S ⊂ R be the set of numbers of the form 1/n, where n is a positive
integer. Find all the limit points of S; is S closed?
Problem 2.
(1) Write down a set S with infinitely many limit points, all of
whose limit points are not in S.
(2) Suppose we have a set S ⊂ Rn with a limit point p ∈ Rn . Show that
the open ball B(p, ) (i.e. points of distance less than from p) contains
infinitely many points of S, for any choice of .
Problem 3. Let f : Rn → R be a continuous function; recall this means that
limx→a f (x) = f (a).
(1) Using the definition of continuity, show that the preimage of any open set in
R is an open subset of Rn (i.e. if U ⊂ R is open, then so is f −1 (U ) ⊂ Rn ).
(2) Using the previous part, show that the preimage of a closed set is closed
(use that closed sets are complements of open sets).
(3) Show that the set {(x, y, z) : −5 < x2 + sin(yz) − 30xyz + ez < 5} is an
open subset of R3 .
Problem 4. Given a convex set S ∈ Rn , show that the interior of S is also a
convex set. (Hint: if x and y are in the interior of S then so are all points close
enough to each of them, say within distance . Now if we have a weighted average
z = tx + (1 − t)y, show that every vector within distance of z is also contained in
S).
Problem 5. In each part, give an example of S ⊂ R and a continuous function
f : S → R such that
(1) S is closed but f (S) is not closed
(2) S is open but f (S) is not open
(3) S is bounded but f (S) is not bounded.
√
Problem 6. Consider the function f (x) = cos( xex ). Show that there exists a
number α and a sequence of positive integers a1 , a2 , a3 , . . . such that
|f (a1 ) − α| < 1, |f (a2 ) − α| < 1/2, |f (a3 ) − α| < 1/3, . . . .
(Hint: remember the definition of sequential compactness and apply it to [−1, 1])
Problem 7. Consider the set
S = {(x, y)| − 2 ≤ −2y 3 + log(1 + |xy|) + 3x4 ≤ 7, x2 + y 2 ≤ 4}.
Show there exists (a, b) ∈ S such that
cos(|x| + ey ) ≤ cos(|a| + eb )
for all (x, y) ∈ S.
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