CHAPTER 1 PRELIMINARIES
1.
1 REAL NUMBERS AND THE REAL LINE
1.
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1, 0.
2, 0.
3, 0.
8, 0.
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2 3 8 9
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2.
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More
CHAPTER 1 PRELIMINARIES
1.
1 REAL NUMBERS AND THE REAL LINE
1.
Executing long division, 0.
1, 0.
2, 0.
3, 0.
8, 0.
9"
9 9 9 9 9
2 3 8 9
œ œ œ œ œ
2.
Executing long division, 0.
09, 0.
18, 0.
27, 0.
81, 0.
99"
11 11 11 11 11
2 3 9 11
œ œ œ œ œ
3.
NT = necessarily true, NNT = Not necessarily true.
Given: 2 < x < 6.
a) NNT.
5 is a counter example.
b) NT.
2 < x < 6 2 2 < x 2 < 6 2 0 < x 2 < 2.
Ê c c c Ê c
c) NT.
2 < x < 6 2/2 < x/2 < 6/2 1 < x < 3.
Ê Ê
d) NT.
2 < x < 6 1/2 > 1/x > 1/6 1/6 < 1/x < 1/2.
Ê Ê
e) NT.
2 < x < 6 1/2 > 1/x > 1/6 1/6 < 1/x < 1/2 6(1/6) < 6(1/x) < 6(1/2) 1 < 6/x < 3.
Ê Ê Ê Ê
f) NT.
2 < x < 6 x < 6 (x 4) < 2 and 2 < x < 6 x > 2 x < 2 x + 4 < 2 (x 4) < 2.
Ê Ê c Ê Ê c c Ê c Ê c c
The pair of inequalities (x 4) < 2 and (x 4) < 2 | x 4 | < 2.
c c c Ê c
g) NT.
2 < x < 6 2 > x > 6 6 < x < 2.
But 2 < 2.
So 6 < x < 2 < 2 or 6 < x < 2.
Ê c c c Ê c c c c c c c c c
h) NT.
2 < x < 6 1(2) > 1(x) < 1(6) 6 < x < 2
Less
