# Minimality exercise from Halmos

The task given in Halmos’s *Naive Set Theory* is to show that
given some subset of a natural number , there is some such that for any other , we have . Intuitively, we know that “” means “”, but since we haven’t defined
order yet in the book, we have to deal just with the set theoretic
properties of the naturalvnumbers. Now how should we go about finding
this ? One way is to look at an
example to see what happens. Suppose we take . Then the “” that we want is , since .