Difference of at most $10$

Sorting. Let $a_1<a_2<\cdots<a_{11}$ be the integers.

Range. $a_{11}-a_1\le 100-1=99$.

Average gap. There are $10$ gaps summing to at most $99$, so some gap is $\le\lfloor 99/10\rfloor=9$.

Conclusion. Two integers differ by at most $9$.