5 Homework - Math 139
Due Oct 6th
Instructor
Oce
Oce hours
Web page
Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm-3:30pm.
www.math.duke.edu/ mauro/teaching.html
Reading: from Reeds textbook: Sections 6.1,3.2
Problems:
2.4: #10
2.6: #1, 3, 9
6.1: #1(a,c), 8
Additional Problem:
1. Let {an } and {bn } be Cauchy sequences. Let {an } {bn } mean that an bn 0 1 .
Prove that is an equivalence relation: {an } {an }; if {an } {bn } then {bn } {an };
if {an } {bn } and {bn } {cn }, then {an } {cn }.
2. Prove that the sum and product of Cauchy sequences is Cauchy.
3. Let [an ] denote the equivalence class of the Cauchy sequence {an }. Given Cauchy
sequences {an } and {bn }, dene the sum and product of the equivalence classes containing
them by
[an ] + [bn ] := [an + bn ]
[an ] [bn ] := [an bn ]
Prove that these rules are well-dened by that showing if {an } {an } and {bn } {bn },
then {an + bn } {an + bn } and {an bn } {an bn }
4. If C denotes the set of equivalence classes of Cauchy sequences, then with the sum
and product operations in 3. C is in fact a eld. Dont try to prove this, but identify
0 and 1 in C and verify that [an ] + 0 = [an ] and [an ] 1 = [an ] for all Cauchy sequences
{an }.(Keep in mind that your choice of 0 (or 1) in your answer will be an equivalence
class of Cauchy sequences. This class may be identied by any Cauchy sequence in it.)
1
Note that this is a dierent notion of equivalence than the one I gave and used in class. It is in fact
an equivalent notion, in the sense that the equivalence classes are the same, but you do not need to show
that (albeit that could be an exercise!). Use the equivalence relation dened here to do this problem.