Math 555/704I Course Outline
Spring 2011 Text : A First Course in Real Analysis Springer, Undergraduate Texts in Mathematics by: Sterling K. Berberian Supplemented with notes.
Professor : Anton R. Schep Office : LeConte 300C Email : schep@math.sc.edu Web
Solutions homework 9. Prove that f (x) = limn fn (x) exists for all x R. Does (fn ) (1) Let fn (x) = converge uniformly to f ? Solution: There are 3 cases: |x| < 1, |x| = 1, and |x| > 1. In case |x| < 1, then x2n 0 as n , which implies that fn (x) 0 as n
Page 188 Problem 6. follows now from
n 1 1 x
Solutions homework 8. dx = 2 n - 2 as n . The divergence of the series
n+1
1 A simpler proof follows from the fact that diverges. n Page 188 Problem 7. From ln x x - 1 it follows that ln(1 + n) (1 + n) - 1 = n.
Solutions homework 7. Page 182 Problem 1. The answer is 2 times the sum of the geometric series with c = 1 , 3 1 so the answer is 2 1- 1 = 3. 3 Page 182 Problem 4. For n 1 we have s2n+2 = s2n + a2n+1 + a2n+2 s2n (since a2n+1 +a2n+2 0), so (s2n ) is an inc
Solutions homework 6. Page 141 Problem 2. If f g, then mk (f ) = infcfw_f (x) : xk-1 x xk mk (g) = infcfw_g(x) : xk-1 x xk and thus sf () sg () for any subdivision . By definition this implies that
b b
f
a a
g.
The corresponding inequalities for the upp
Solutions homework 5. Page 128 Problem 3: Using the substitution y = -x we get g(y) - g(-c) f (-y) - f (c) f (x) - f (c) = =- . y - (-c) y - (-c) x-c Now letting y -c- is the same as letting x c+ , from which the problem follows. Page 128 Problem 4. As g(
Solutions homework 4. Page 108 Problem 4: Let (xn ) be a Cauchy sequence and let > 0. Then there exists > 0 such that |x - y| < implies |f (x) - f (y)| < . For this there exist N such that |xn - xm | < for all n, m N . Hence |f (xn ) - f (xm )| < for all
Solutions homework 3. Page 69 Problem 10: a. Assume A and B are neigborhoods of x. Then there exist r1 > 0 such that Ur1 (x) A and r2 > 0 such that Ur2 (x) B. let r = mincfw_r1 , r2 . Then r > 0 and Ur A B. Hence A B is a neighborhood of x. b. Let Br (c)
Solutions homework 2. Page 32 Problem 6: Clearly d(x, y) 0 and d(x, y) = 0 if and only if x = y. Hence property (i) holds. It is also clear that d(x, y) = d(y, x), so it remains to show that the triangle inequality holds. Let x, y, z X. If x = y, then d(x
Solutions homework 1. (1) Prove that [-1; 1) is not compact by using the definition of a compact set (to get credit for the problem, use the definition and not any theorems about compact sets).
1 Proof: Let On = (-1, 1 - n ). Then [-1; 1) n On , but [-1;
Homework 9. Prove that f (x) = limn fn (x) exists for all x R. Does (fn ) (1) Let fn (x) = converge uniformly to f ?
x2n . 1+x2n
(2) Define fn : [0, 1] [0, 1] by fn (x) = xn (1 - x). Prove that fn converges uniformly to 0. (3) Prove that nx + sin(nx2 ) n
Extra problems Homework 7. (1) Let f : [0, 1 R be a continuous function. Prove that 1 lim n n
n 1
f (k/n) =
k=1 0
f (x) dx.
(Hint: Use uniform continuity to show that for > 0 there exists a > 0 such that S() - s() < for all subdivisions with norm N () < .
Extra problems Homework 4. (1) Let f, g : R R be uniformly continuous functions. Assume that both f and g are bounded. Prove that the product f g is uniformly continuous. (2) A function f : R R is periodic, if there exists a c R such that f (x + c) = f (x
Homework 3, Additional Problems. (1) Let (X, d) be a metric space. a. Let Ei X (i cfw_1, , n) be a finite collection of subsets of X. Prove that n Ei = n Ei . i=1 i=1 b. Let Ei (i I) now be an arbitrary collection of subsets of X. Prove that iI Ei iI Ei a
Homework 1. (1) Prove that [-1, 1) is not compact by using the definition of a compact set (to get credit for the problem, use the definition and not any theorems about compact sets). (2) What is an interior point? Prove that
1 4
is an interior point of (
Math 555/704I Course Outline
Spring 2011 Text : A First Course in Real Analysis Springer, Undergraduate Texts in Mathematics by: Sterling K. Berberian Supplemented with notes.
Professor : Anton R. Schep Office : LeConte 300C Email : schep@math.sc.edu Web