I'm trying to define some quantity as the maximum natural number $N$ that allows the truth of an inequality. The way I've been thinking of writing it would be:
$$N = \max_{n \in \mathbb{N}} (f(n) < c)$$
$f(N)$ would be a function of $N$. Is this correct? I haven't seen any example of someone doing this.
Thank you for your help!
In general, the notation $$\max_{x \in S}(f(x))$$ means that we maximize the quantity $f(x)$ over all $x \in S$. Using it in this case does not make sense, since $f(n) < c$ is not a quantity we can maximize.
Instead, we can begin by defining the set $\{n \in \mathbb N : f(n) < c\}$, or $\{n \in \mathbb N \mid f(n) < c\}$: the set of all $n \in \mathbb N$ that satisfy the condition. Then $$\max\{n \in \mathbb N : f(n)<c\}$$ will be the maximum element of this set, which is the largest natural number satisfying the condition $f(n)<c$.
(This is a slightly different version of the $\max$ operator: for a set $S$, $\max S$ is equivalent to $\displaystyle \max_{x \in S}(x)$.)