Hi i am reading about lesser know construction of real numbers by Kolmogorov. In his construction real numbers are defined as a set $\Phi$ of functions $\alpha: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy two following properties.
Property 1: For any $k, n \in \mathbb{N}$ following inequality holds
$$ k\alpha(n) \leq \alpha(kn) < k(\alpha(n) + 1). $$
Property 2: For any $n \in \mathbb{N}$ there is $k \in \mathbb{N}$ such that $$ k\alpha(n) < \alpha(kn). $$
My problem is that i have no idea how to prove that set $\Phi$ has greater cardinality than $\mathbb{N}$. Proof that set of all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ is not countable is easy. But how to prove that set of all such functions that also satisfy properties 1 and 2 is also uncountable?
For people interested in construction itself here is the link. (in Russian)
