Math 5070 Fall 2012 Homework 7 solutions
1. 44.5 Let M1 , M2 , M3 be metric spaces and g : M1 M2 , and f : M2
M3 uniformly continous. Prove that f g is uniformly continuous.
Solution. Denote d1 , d2 , d3 the metrics on M1 , M2 , M3 , respectively. Let
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Math 5070 Fall 2012 Homework 3 solutions
1. 35.4 Let M be a set and d be a function from M M into R which
satises the three properties
(i) d(x, y ) = 0 if and only if x = y,
(ii) d (x, y ) = d(y, x) for all x, y M,
(iii) d (x, z ) d (x, y ) + d (y, z ) fo
Math 5070 Fall 2012 Homework 2 solutions
1. 20.8 Let cfw_an be a bounded sequence. Prove that there exists subsequences cfw_ank and cfw_amk of cfw_an such that
lim ank = lim sup an
k
lim amk = lim inf an
k
n
n
Solution. Denote s = lim supn an . From p
Math 5070 Fall 2012 Homework 1 solutions
1. 5.7 Let X and Y be sets of real numbers with least upper bounds a and
b, respectively. Prove that a + b is the least upper bound of he set X + Y =
cfw_x + y |x X, y Y .
Solution. Because a = sup X, a is upper b
NAME: .
Math 5070 Fall 2012 Midterm October 18, 2012
No books or notes are allowed except for the use of one handwritten lettersized sheet (both sides). Please attach the sheet to the exam.
Please do not forget to put your name on the exam and number all
Math 5070 Fall 2012 Final December 11, 2012 5:00-7:00 Room CU 656
No books or notes are allowed except for the use of one handwritten lettersized sheet (both sides). Please attach the sheet to the exam. Please do not forget
to put your name on the exam, n
Math 5070 Fall 2012 Homework 4 solutions
1. 38.5 Let X be subseet of a metric space M . We say that a point x in
M is an accumulation point of X there exists a sequence cfw_xn in X such that
limx xn = x and xn = x for all n. Denote by X a the set of all
Math 5070 Fall 2012 Homework 5 solutions
1. 39.9 Let X M , M a metric space. Prove that X is open in M if and
only if X is the union of open balls.
Solution. Let X be open in M . Since X is open,
x X (x) > 0 : B(x) (x) X.
Since x B(x) (x),
cfw_x
X=
x X
B
Math 5070 Fall 2012 Homework 10 solutions
1.
60.10 Prove that C [a, b] is a complete metric space with the metric d (f, g ) =
sup cfw_|f (x) g (x)| : x [a, b].
Proof. Suppose cfw_fn is Cauchy in C [a, b]. Then cfw_fn is uniformly Cauchy and thus converg
Math 5070 Fall 2012 Homework 11 solution
1. 64.2. Give an example of a divergent series n=0 an such that n=0 an xn
converges if |x| < 1 and limx1 n=0 an xn exists.
Solution. The series needs to have radius of convergence R = 1 otherwise the
series n=0 an
Math 5070 Fall 2012 Homework 9 solutions
1. Let fn , f : M1 \ cfw_a M2 where (M1 , d1 ) and (M2 , d2 ) are metric spaces, (M2 , d2 ) is complete, and
a M1 . Prove that if fn
f on M1 \ cfw_a and limxa fn (x) = bn exists for all n, then limn bn = b
exists,
Math 5070 Fall 2012 Homework 6 solutions
1. 41.4 Let M be a metric space and X M with the relative metric. Prove
that if f is a continous function on M , then f |X is a continuous function on X .
Solution. Let limn xn = x in X . Then limn xn = x in M . Si
Math 5070 Fall 2012 Midterm October 18, 2012
1. Let (M, d) be a metric space and X M . Suppose the metric space (X, d ) with the reduced
metric is complete. Show that X is closed in (M, d) .
Solution. Let cfw_xn X and limn xn = x in M . Since cfw_xn is