$D=\{X\subseteq \mathbb{N}:|\mathbb{N}\setminus X| < |\mathbb{N}|\}$
I would like to show that $|D|=|\mathbb{N}|$. I know Cantor-Berenstein theorem. When it comes to lower bound:
Let's consider following sets $(\mathbb{N}\setminus \{1\}), (\mathbb{N}\setminus \{2\}), (\mathbb{N}\setminus \{3\}),..$
It is obvious that $Y=|\{\mathbb{N}\setminus \{n\}: n\in\mathbb{N}\}|=\aleph_{0}$. $Y\subseteq D$ hence, $|D|\ge \aleph_{0}$.
I can't show that $|D|\le \aleph_{0} $. Could you help me ?
