Complex sentence: Since neither of these sets is finite, neither set is in the Fréchet filter on N. However, the Fréchet filter is included in every free filter, and an ultrafilter is free if and only if it includes the Fréchet filter.
Split into simpler sentences: Since neither of these sets is finite, neither set is in the Fréchet filter on N. However, the Fréchet filter is included in every free filter. Moreover, an ultrafilter is free if and only if it includes the Fréchet filter.