Cardinality of set of well-ordered sequences

Question

We think of $A=\mathbb{R}^\mathbb{N}$ as the set of all functions $f:\mathbb{N}\to\mathbb{R}$. Consider the following subset of $A$: $$ B=\{f\in A\mid f(\mathbb{N}) \text{ is a well-ordered subset of $\mathbb{R}$ w.r.t. the standard ordering of $\mathbb{R}$}\}. $$ I'm wondering what the cardinality of $B$ is. Any help would be much appreciated.

Answer 1

Note that $|A|=|\mathbb{R}|^{|\mathbb{N}|}=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$, so we have $|B|\leqslant 2^{\aleph_0}$. On the other hand, for every $r\in(-\infty,0)$, the map $f_r\in A$ that sends $0$ to $r$ and sends $n$ to $n$ for all $n>0$ is an element of $B$. (Why?) The association $r\mapsto f_r$ is an injection $\mathbb{R}\to B$, so we have $|B|\geqslant 2^{\aleph_0}$, and hence $|B|=2^{\aleph_0}$.