In other words, every internal infinite subset of the hyperreals necessarily contains non-standard elements.

A -array is a Borel function for some infinite subset of .

An infinite subset of the natural numbers cannot be bounded from above.

Nevertheless, there are (many) non-standard whole numbers in any infinite subset of N.

We denote by the set of all infinite subsets of .

An infinite set may contain an infinity of infinite subsets.

There exists a set such that every infinite subset of has precisely seven elements.

Since no infinite subset of any set has precisely seven elements, we may be tempted to conclude that this statement is obviously false.

So, there must be an infinite subset of all i's which belongs to same class.

For each of these sets, there were infinite subsets.