# Negative association in a “k out of n” process

Suppose we have $$n$$ distinct balls labeled $$1,2,\dots,n$$ in a black box. Now we want to fetch $$k$$ balls from the box, one by one. Let event $$E_i$$ be that the label of the $$i$$-th ball we fetch is no more than $$m$$, where $$m \leq n$$.

By intuition I think events $$E_1, E_2, \dots, E_k$$ are negatively associated. However after some simple tries I feel that I cannot easily make a proof. Since this process is quite natural I believe it has been studied before or it is quite easy to people who are familiar with probability theory.

Question. Can we prove that $$E_1, E_2, \dots, E_k$$ are negatively associated?

This question arises in my research, in which I need to solve it to prove that my algorithm behaves as desired. However I am worried that it may not be a research-level question because of its simple form. Let me know if it is not appropriate to put this question here.