Given a number of different fractions, create a new fraction whose numerator is the sum of all those fractions' numerators and whose denominator is the sum of the denominators. Call it y. Call the smallest of the original fractions x and the largest z.
Prove that for all cases, x < y < z.
Suppose we have a1
< ... < an
. These imply that a1
for all i, with equality iff i=1. Adding the inequalities yields a1
), or a1
), which proves the first inequality. The second one follows analogous.