On pg 76 of Jech's Set Theory, he proves the existence of a nonprincipal ultrafilter on $\omega$ that is not a $p$-point. 



> 
> Given a partition $\{A\_n\}$ of $\omega$ into $\aleph\_0$ infinite pieces, let $F$ be the following filter on $\omega$.  
> 
> $X \in F$ if and only if except for finitely many $n$, $X \cap A\_n$ contains all
>  but finitely many elements of $A\_n$.  
> 
>  If $D$ is any ultrafilter extending $F$, then $D$ is not a $p$-point.
> 
> 
> 


I've been thinking why the ultrafilter $D$ isn't a $p$-point but I couldn't figure it out 



A nonprincipal ultrafilter that is not a $p$-point
First of all, it is clear that the given Partition $\{A\_n\;|\;n\in\omega\}$ satisfies $A\_n\notin D$ for any n, since $\omega\smallsetminus A\_n\in F\subseteq D$ for any n (since $(\omega\smallsetminus A\_n)\cap A\_k=A\_k$ for all $k\neq n$).


Furthermore, assume $X$ has the property that $|X\cap A\_n|<\omega$ for any $n$. Then
$$|A\_n\cap(\omega\smallsetminus X)|=|A\_n\smallsetminus X|=|A\_n\smallsetminus(A\_n\cap X)|=\omega$$
And therefore $\omega\smallsetminus X\in F$ and $X\notin D$.


