I have a poset $(\mathcal{P}(\mathbb{N}),\subseteq)$, where $\mathcal{P}$ is a power set, and I need to find a non-empty chain, such as that chain would not have the least or greatest element.
It is easy for me to imagine a chain in $(\mathcal{P}(\mathbb{N}),\subseteq)$, but I can't really grasp the idea of how do I make up a chain with no least element.
From what I have googled so far, I can suppose it should be somehow related to the poset $(\mathbb{Q},\subseteq)$, but I don't know how to develop that idea (in case it shall lead me to an answer at all).
I. Define $A_n\in\mathcal P(\mathbb N)$ for $n\in\mathbb Z$ by setting $$A_n=\{1,3,5,7,\dots\}\setminus\{1,3,5,\dots,2|n|-1\}\text{ for }n\lt0,$$ $$A_n=\{1,3,5,7,\dots\}\cup\{2,4,6,\dots,2n\}\text{ for }n\ge0.$$ Then $m\lt n\implies A_m\subsetneq A_n$.
II. Establish a one-to-one correspondence netween $\mathbb N$ and the set $S=\{x+iy:x,y\in\mathbb N\}$. Define a chain in $\mathcal P(S)$ by setting $B_\theta=\{z\in S:0\lt\arg z\lt\theta\}$ for $0\lt\theta\lt\frac\pi2$.
Then $\phi\lt\theta\implies B_\phi\subsetneq B_\theta$.