Math 23b
Homework 7
Spring, 2009
Due Wednesday, April 8.
Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/,
w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the
problem does not explicitly ask you to do so. In addition, one third of the grade on each
exercise will be determined by the presentation of your argument.
Even if your answer
is in the end correct, you will lose points if there are irrelevant, extraneous or incorrect
statements in your argument. There will be no revision for this homework. Since there
is restriction on how much you can write in each problem, you should consider carefully
what is essential to include, before writing your final answers.
1. Let
y > x
.
(a) In 1 line, prove that there is a positive integer
n
such that
n
(
y

x
)
>
1.
(Use the
Archimedean property.)
(b) In no more than 10 lines, prove that there is an integer
m
such that
m > nx
≥
m

1.
(Consider the case
nx >
0 and
nx
≤
0 separately. Use the Archimedean property to find
one
m
, and then use the Well Ordering property to find the smallest
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 Spring '09
 BongLian
 Math, lim, Archimedean Property, Archimedean

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