Twenty distinct integers are chosen from $\{1,2,...,69\}$. Prove that amongst their pairwise differences there are at least four which are identical.
I understand that the set $\{1...69\}$ is arbitrary. I'm having a hard time proving it. Should I prove through induction or use the pigeon hole principle?
Let the integers be $a_1\lt a_2\lt\cdots\lt a_{20}$, and let $b_1,b_2,\dots,b_{19}$ be given by $b_i=a_{i+1}-a_i$. Then $\sum_1^{19}b_i\le68$. But if you have $19$ positive integers, no one integer more than three times, then their sum must be at least $(1+1+1)+(2+2+2)+\cdots+(6+6+6)+7=70$. So some difference occurs at least $4$ times.