Posted by Anonymous on .
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?

Math Word Problem 
MathMate,
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.