"If a linearly ordered set $P$ has a countable dense subset $D$, then $|P|\le2^N$"
1) Is the set $D$ dense in $P$ ?
2) I know there is another link regarding the same question but I do not see how the solutions provided form an injection from $P$ to $2^N$(or $P(N)$ or R or ...).
A linearly ordered set $P$ has a countable dense subset, then $|P|\le 2^N$
3) My initial solution was to consider the sets $\forall p\in P$ $S_p = \{x\in D | x < p\}$ but I still can't get any useful injection.
Any hints or suggestions on how to proceed?
If $p\ne q$ then $S_p\ne S_q$.
We may assume wlog. that $p<q$. Then there exists $x\in D$ with $p<x<q$. But that means $x\in S_q$ and $x\notin S_p$.