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Unformatted text preview: Modern Analysis, Homework 1
Due June 3, 2010 1.
2.
3.
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5.
6. (1 pt) Let k be a ﬁeld and x ∈ k be nonzero. Show that 1/(1/x) = x.
(3 pts) Rudin, Ch 1, problem 5.
(5 pts) Rudin, Ch 1, problem 6.
(10 pts) Rudin, Ch 1, problem 7.
(2 pts) Rudin, Ch 1, problem 15.
Consider the set of inﬁnite decimals, R. That is, expressions of the form
±an an−1 · · · a2 a1 .b1 b2 · · · for some n ∈ N and each ai , bi ∈ {0, 1, 2, · · · , 9}. Also stipulate that if an expression ends
with inﬁnitely many 9’s, then that is the same expression with those 9’s replaced by 0’s and
the last non 9 entry incremented by one. For instance, 1.299999 · · · = 1.3. Therefore we may
assume that any decimal does not end in inﬁnitely many 9’s.
Given x ∈ R, deﬁne x[n] as the nth truncation of x, ie change x by replacing each decimal at
distance at least n +1 to the right of the decimal point to 0. For instance, if x = 3.1415926 · · · ,
then x[3] = 3.141 (yeah yeah, inﬁnitely many 0’s, who cares).
Given decimals x, y ∈ R, deﬁne x < y to mean x[n] < y [n] for some n (here we are using the
< on Q).
(a) (2 pts) Explain why this is is well deﬁned (ie independent of n) and deﬁnes a total order
on R.
(b) (5 pts) Show that (R, <) satisﬁes the leastupperbound property.
(c) (Bonus, not necessary) Deﬁne addition on R. Then deﬁne multiplication. 1 ...
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters

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