Posted by **Anonymous** on Friday, October 9, 2009 at 8:10pm.

Let A be the set containing all rational numbers that are less than 5. Is there a rational number q in set A such that all other numbers in set A are less than q? Why or why not?

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**MathMate**, Friday, October 9, 2009 at 8:47pm
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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