Formalizing the Mathematical statement in Graph theory This question is in Diestel - Graph Theory (5th Edition) Exercise 1.15.

Let $\alpha, \beta$ be two graph invariants with positive integer values. 
   Formalize the two statements below, and show that each implies the other:
(i) $\beta$ is bounded above by a function of $\alpha$;
(ii) $\alpha$ can be forced up by making $\beta$ large enough.

I have trouble formalizing statement (ii). I have no idea how to formalize the word "forced up".
 A: Formalize as follows:
(i)
There exists a function $f: \mathbb{N} \to \mathbb{N}$ such that $\beta(G) \le f(\alpha(G))$ for every graph $G$.
(ii)
For every graph $G$ there is a positive integer $N$ such that for all graphs $G'$ with $\beta(G') > N$ we have $\alpha(G') > \alpha(G)$.
Suppose (i) holds.
Let $G$ be any graph.
Then $\beta(G) \le f(\alpha(G))$.
Choose
$$N = \max\{f(0), f(1), \dots, f(\beta(G))\}.$$
Now let $G'$ be a graph such that $\beta(G') > N$.
Then $N < \beta(G') \le f(\alpha(G'))$, which implies that $\alpha(G') \notin \{0, 1, \dots, \beta(G)\}$.
Hence $\alpha(G') > \alpha(G)$.
Conversely, suppose (ii) is true.
Define $L(n) = \{G: \alpha(G) = n\}$.
We need to find a function $f: \mathbb{N} \to \mathbb{N}$ such that, for all $n \in \mathbb{N}$, we have $f(n) \ge \beta(G)$ for all $G \in L(n)$ (in case $L(n) = \varnothing$ for some $n$, $f(n)$ can be anything).
It suffices to show that $\beta$ has a maximum on each nonempty $L(n)$.
Fix an $n \in \mathbb{N}$ for which $L(n)$ is not empty.
There exists an $N_G \in \mathbb{N}$ such that, for all graphs $G'$ with $\beta(G') > N$, we have $\alpha(G') > n$.
This implies that $\beta(G) \le N$ for all $G \in L(n)$, completing the proof.
The function could be $f(n) = N_G$ for some G \in L(n)
