Let $a_i$, $1 \leq i \leq 5$ denote five positive real numbers such that $\sum_{i =1}^{5}a_i = 100$. Show that there exist a pair $a_i,a_j$ such that $|a_i-a_j|\leq 10$. Is there a proof using pigeon hole principle. I think i have a proof but it does not involve pigeon hole principle. My proof is:
Suppose there exist no such pair $a_i,a_j$. Assume that $a_1 < a_2 < a_3 < a_4 < a_5$, then $a_1 \in [0,20]$. As $a_i > a_{i-1} +10$ $ \forall 1 \leq i \leq 4$, we have $100 = \sum_{i=1}^{5}a_i > 5a_i +100$. Hence by contradiction the claim cannot be true.
It is easy to turn your proof to a pigeon hole principle argument.
Let $a_1<a_2<a_3<a_4<_5$, then
$$100=(a_5-a_4)+2(a_4-a_3)+3(a_3-a_2)+4(a_2-a_1)+5a_1$$
Applying pigeon hole principle to $(a_5-a_4)$, 2 copies $(a_4-a_3)$, 3 copies $(a_3-a_2)$, 4 copies $(a_2-a_1)$.