Given the lattice $(X,\cup,\cap)$ of all finite and cofinite subsets of $P(\mathbb{N})$ I have to prove, that this is not a complete lattice.
At first I thought I take the union of all subsets of $X$ which have a single, even element. The resulting set would be the set of all even numbers. This set is not in $X$ because it is neither finite nor cofinite.
But then I realized that I can't do that because this set was never a subset of $X$ in the first place.
So, does there even exist a subset of $X$ which has no join or meet in $X$?
Let $E=\big\{\{2n\}:n\in\Bbb N\big\}$, the set of all singletons of even natural numbers. Clearly $E\subseteq X$, but $E$ has no supremum in $X$: every set of the form $\Bbb N\setminus F$ such that $F$ is a finite set of odd natural numbers is an upper bound for $E$, and none of these upper bounds is minimal.