Math 410
Spring 2016
Review
Give a critique of each of the following arguments. If the argument is correct,
fill in any missing details; if the argument is incorrect, explain what part(s)
of the argument are incorrect.
1. Let A and B be non-empty bounded
Math 410
Spring 2016
Problems January 22
1. Let x, y, z 2 Q. Prove that axioms we used to construct Q imply the
following statements. (For (b), (c), and (d), you may use (a) in your
proof.)
(a) * If x 6= 0 and xy = xz, then y = z.
(b) If x 6= 0 and xy = x
Math 410
Spring 2016
Problems March 25
19. Let a, b R with a < b. Construct an open cover of (a, b) that DOES
NOT contain a nite subcover.
20. (a) Let K1 and K2 be compact sets. Prove that K1 K2 is compact.
(b) If K1 , K2 , K3 , . . . are each compact, is
Advanced Calculus I
Math 410, Spring 2016
Course Times: Monday, Wednesday, Friday 11:15 am to 12:05 pm, Burruss 0034
Professor: John Webb Email: [email protected]
Office: Roop 326
Phone: 540.568.5159
Office Hours: Whenever my door is open, I welcome you to c
Math 410
Spring 2016
Problems April 1
23. For each given sequence cfw_an
n=1 , limit a, and value of , nd a value
for N such that for all n N , the term an is within of a.
1
(a) cfw_an
n=1 = cfw_ n2 n=1 , a = 0, and = .001.
1/n
(b) cfw_an
n=1 , a = 1
Math 410
Spring 2016
Problems February 22
12. Briefly justify your answer to each question belowyou do not need to
give a formal proof.
(a) Is Q an open subset of R?
(b) Is Q a closed subset of R?
(c) Is every q 2 Q a limit point of Q?
13. Construct an ex
Math 410
Spring 2016
Problems February 5
5. For each part, nd an example of a set that meets the given conditions.
(a) A set A with lub(A) = 1, glb(A) = 0 with 1 A and 0 A.
(b) A set B with lub(B) = 1, glb(B) = 0, and .5 B.
(c) A set C that is bounded and
Math 410
Spring 2016
Hint for #28(c)
Let > 0. Then the goal of this problem is to show that there exists
some k N such that for any n1 , n2 N with n1 , n2 k, we have
|sn1 sn2 | < .
Since cfw_an
n=1 is Cauchy, there exists some N N such that for any m1 ,
Math 410
Spring 2016
Test 1 Real Topology Review
1. (a) Let A and B be open sets of R. Prove that A B and A B are
both open sets.
S
(b) T
Let A1 , A2 , . . . , An be open sets of R. Prove that ni=1 Ai and
n
i=1 An are both open sets.
S
(c) Let A1 , A2 , .
Math 410
Spring 2016
Problems March 18
17. Write a formal proof for each of the statements below.
(a) Let x R satisfy the condition that for any y R, xy = x. Then
x = 0.
(b) Let B = cfw_ r : r Q, r > 0. Then card(B) = card(N).
(c) Let A R be bounded above
Math 410
Spring 2016
Theorem: A continuous function on a compact set is uniformly continuous.
Proof. Let f : K R be continuous with K a compact set, and let > 0.
For each x K define Gx = (f (x) /2, f (x) + /2). Since Gx is open, then
f 1 (Gx ) is an open