Is the intersection of a decreasing (w.r.t. inclusion) sequence of cofinal subsets again a cofinal subset?

Question

Let $(D, \le )$ be a directed set. A subset $A$ of $D$ is cofinal in $D$ if and only if $\forall d \in D, \exists a\in A, d \le a$. Let $(D_n)$ be a decreasing (w.r.t. inclusion) sequence of cofinal subsets of $D$.

Is it true that $\bigcap_{n\in \mathbb N} D_n$ is cofinal in $D$?

Answer 1

As @Geoffrey Trang pointed in a comment, such intersection could be an empty set. For example, $D=\mathbb{N}$ and $D_n=\{m \in \mathbb{N} \mid m \ge n\}$.