The definitions that I am using are:
:
A cofinal ultrafilter $\mathcal {U} $ on a directed set $I $ is an ultrafilter which contains the cofinal filter base, i.e. family of all subsets of the form $\{i\in I: i\geq i_0\} $.
I thought that it is non principal or contains no finite sets at least in the case that $I $ is infinite. However, I succeded to do so just under the assumption that there is no $y\in I $ such that $y\geq x $ for all $x\in I $.