why is the average of a set of numbers alway smaller than the largest number in the set?
I assume smaller is in the weak sense rather than in a strict sense.
Let us do it with three elements so you can see the idea. Say $\displaystyle \{a_1\leq a_2\leq a_3\}$.
The average $\displaystyle a_1=\tfrac{a_1+a_1+a_1}{3}\leq \tfrac{a_1+a_2+a_3}{3} \leq \tfrac{a_3+a_3+a_3}{3} = a_3$