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Tuesday, November 06, 2012

 

Why rationals are incomplete

Let A = {rQ | r>0r2>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:
  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
  x=pq where p,qN and x2>3
So
  p2>3q2
Let's consider the rational number
  n22n+1 where nN
which increases without bound as n increases. In particular, we can always pick a large enough n so that
  n22n+1>3q2p23q2n2p23n2q2>6q2n+3q2n2p2>3q2(n+2n+1)n2p2>3q2(n+1)2n2(n+1)2p2q2>3
Let
  y=n(n+1)pq
Since n<n+1,
  p2q2>n2(n+1)2p2q2>3x2>y2>3
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
  x=pq where p,qN and x2<3
So
  p2<3q2
Let's consider the rational number
  n22n+1 where nN
which increases without bound as n increases. In particular, we can always pick a large enough n so that
  n22n+1>p23q2p23n2q2n2p2>2p2n+p23n2q2>p2(n+2n+1)3n2q2>p2(n+1)23>p2q2(n+1)2n2
Let
  y=pq(n+1)n
Since n<n+1,
  3>p2q2(n+1)2n2>p2q23>y2>x2
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.


Comments:
You forgot to mention p-adic norms.
 
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