Posted by Anonymous on Friday, October 9, 2009 at 8:10pm.
We will try to prove this by contradiction.
Hypothesis: existence of q=a/b which is the largest rational number less than 5, i.e. q=a/b<5 and b≠0, and that no other rational number exists that is larger than q and less than 5.
We will calculate the r, the average between q and 5
r=(q+5)/2
=(a/b+5)/2
=(a+10b)/2b
so that
q<r<5
which means that
r is greater than q,
r is less than 5, and
r is rational.
Therefore that hypothesis that q exists is false.
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