The question is from chapter 2, problem 5 in Stein's book 'Real analysis':
There is an ordering $≺$ of $\mathbb R$ with the property that for each $y\in\mathbb R$ the set $\{x\in\mathbb R : x ≺ y\}$ is at most countable. The existence of this ordering depends on the continuum hypothesis, which asserts: whenever $S$ is an infinite subset of $\mathbb R$, then either $S$ is countable, or $S$ has the cardinality of $\mathbb R$ (that is, can be mapped bijectively to $\mathbb R$.)
[Hint: Let $≺$ denote a well-ordering of $\mathbb R$, and define the set $X$ by $X = \{y \in \mathbb R \mid \{x \mid x ≺ y\} \text{ is not countable}\}$. If $X$ is empty we are done. Otherwise, consider the smallest element $y \in X$, and use the continuum hypothesis.]
I cannot figure out how this hint will work?
HINT: Let $y$ be the smallest element in $X$. Then $\{x\in\Bbb R:x\prec y\}$ is uncountable, so by the continuum hypothesis it must have the same cardinality as $\Bbb R$. Therefore there is a certain very nice kind of function between $\{x\in\Bbb R:x\prec y\}$ and $\Bbb R$ that you can use to define an ordering of $\Bbb R$ with the desired property.