How can we prove the following: if a relation $R$ is a weak order on a set $X$ and $X_\sim$ is finite, then there exist a function $v:X\to \mathbb{N}$, which is a value representation of preference relation $R$.
Here I know the construction of the function $v$ when the range is the set of real numbers (as $X_\sim$ is countable). But,I am confused when the range of $v$ is the set of natural numbers.
Could anyone please help me?