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Unformatted text preview: Homework 5  Math 139
Due Oct 6th Instructor
Oﬃce
Oﬃce hours
Web page Mauro Maggioni
293 Physics Bldg.
Monday 1:30pm3:30pm.
www.math.duke.edu/˜ mauro/teaching.html Reading: from Reed’s 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 }, deﬁne the sum and product of the equivalence classes containing
them by
[an ] + [bn ] := [an + bn ]
[an ] · [bn ] := [an bn ]
Prove that these rules are welldeﬁned by showing that if {an } ∼ {a′n } and {bn } ∼ {b′n },
then {an + bn } ∼ {a′n + b′n } and {an bn } ∼ {a′n b′n }
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 ﬁeld. Don’t 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 identiﬁed by any Cauchy sequence in it.) 1 Note that this is a diﬀerent 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 deﬁned here to do this problem. ...
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This note was uploaded on 01/16/2012 for the course MATH 139 taught by Professor Pardon,w during the Fall '08 term at Duke.
 Fall '08
 Pardon,W
 Math

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