Let A = {r∈Q | r>0∧r2>3}. Show that A has a lower bound in Q but no greatest lower bound in Q. Give all details of the proof.
== Attempt ==
To show that A has a lower bound in Q is trivial. Clearly, zero is a lower bound of A in Q, as 0 < r for all r in A, and 0 is rational.
Let's assume A has a greatest lower bound x in Q. There are then only three possibilities:
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Case 1: x2=3Case 2: x2>3Case 3: x2<3 |
Case 1: x2=3
The first case is obviously not possible, since √3 is irrational. See proof here.
Case 2: x2>3
For the second case, suppose
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x=pq where p,q∈N and x2>3
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So
Let's consider the rational number
which increases without bound as n increases. In particular, we can always pick a large enough n so that
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n22n+1>3q2p2−3q2n2p2−3n2q2>6q2n+3q2n2p2>3q2(n+2n+1)n2p2>3q2(n+1)2n2(n+1)2⋅p2q2>3
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Let
Since
n<n+1,
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p2q2>n2(n+1)2⋅p2q2>3x2>y2>3
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This means y is an element of A but less than x, which contradicts the assumption that x is a lower bound of A in
Q ! So the supposition that
x2>3 must be false.
Case 3: x2<3
For the third case, suppose
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x=pq where p,q∈N and x2<3
|
So
Let's consider the rational number
which increases without bound as n increases. In particular, we can always pick a large enough n so that
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n22n+1>p23q2−p23n2q2−n2p2>2p2n+p23n2q2>p2(n+2n+1)3n2q2>p2(n+1)23>p2q2⋅(n+1)2n2
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Let
Since
n<n+1,
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3>p2q2⋅(n+1)2n2>p2q23>y2>x2
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This means y is a lower bound of A in
Q but greater than x, which contradicts the assumption that x is the greatest lower bound of A in
Q ! So the supposition that
x2<3 must be false.
As shown in all cases above, it is not possible for A to have a greatest lower bound in Q. This completes the proof.
# posted by rot13(Unafba Pune) @ 10:08 PM
