Homework 3 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
17.4. Let cfw_an be a sequence with positive terms such that limn an = L > 0. Let x be a re
Homework 1 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
2.2. Let h : X Y , g : Y Z , and f : Z W . Prove that (f g ) h = f (g h).
Solution. Let x X
Homework 2 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
9.6. Prove that if A and B are countable sets, then A B is countable.
Solution. For a xed a
Homework 4 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
n=1 bn
26.5. Let n=1 an and
converges.
be absolutely convergent series. Prove that the serie
Homework 5 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
29.15. Let a1 = 1 and an = 1/2n1 for n 2. Let cfw_bm,n
a1 a1 a1 a1
0 a2 a2 a2
0 0 a3 a3
0
Homework 6 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
38.6. Let f be a continuous function from R to R. Prove that cfw_x : f (x) = 0 is a closed s
Homework 7 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
42.1. Prove that none of the spaces Rn , l1 , l2 , c0 , or l is compact.
Solution. Let X = R
Homework 8 Solutions
Math 171, Spring 2010
Henry Adams
The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
44.2. (a) Prove that f (x) = x is uniformly continuous on [0, ).
(b) Prove that f (x) = x3 i