Define the Schnirelmann density of a set of integers by $\sigma(A) = \inf_{n \in \mathbb N} \frac{A(n)}{n}$ where $A(n)$ is the number of elements of $A$ which are $\le n$.
I would like a proof that there is a subset $A'$ of $A$ with the same density and the additional property that removing any element of $A'$ would decrease its density. I could not come up with one myself. Thanks for any proof or sources.