prove by contradiction: there is no greatest negative rational number?

2 Answers

• You basically have it already in your update. You just have to write it down a little more formally.

  Assume that there is a greatest negative rational number r.

  The number is of the form r = -p/q, where p and q are positive integers.

  Divide that number by 2 to get -p/2q.

  That new number is rational and negative since 2q is a positive integer and that doesn't affect the sign or the rationality.

  But let's compare -p/q and -p/2q.

  Take the absolute value of each:

  |-p/q| = p/q

  |-p/2q| = p/2q

  If we compare these we see:

  1(p/q) > ½(p/q)

  1 > ½

  The absolute value is another way of describing the distance from the number to zero. Since -p/q is further from zero and they are both negative, that means -p/2q is closer to zero and thus greater.

  But this contradicts our initial statement that r = -p/q is the largest negative rational number.

  QED: There is no negative rational number greater than all other negative rational numbers.

• Easy. Let X be that greatest negative rational number. But then (1/2)x is greater, and is also rational. Contradicting the assumption.