I have a sequence of ordered pairs: $$(0,1), (0,2), (0,3), (0,4), \ldots, (0, n),$$ $$(1,2), (1,3), (1,4), \ldots, (1, n),$$ $$(2,3), (2,4), \ldots, (2, n),$$ $$(3,4), \ldots, (3, n),$$ $$\ldots,$$ $$(m,n),$$ where $m<n$ $\forall m, n \in \mathbb{N}_0$ and the maximal value of $m$ equals $n-1$.
Question. How to write down this sequence with the set notation?
Edit. My attempt is: $A=\{(m,n): m<n, max(m)=n-1, m,n \in \mathbb{N}_0 \}$.
I suspect that what you want is : $A_n=\{(i,j)\in\mathbb N^2\mid 0\le i\le n-1,\ i+1\le j\le n\}$
where the last couple is $(n-1,n)$ and $m=n-1$
Can be shortened to $A_n=\{(i,j)\in\mathbb N^2\mid 1\le i+1\le j\le n\}$
Note that $A_n$ is finite and depends on the value of $n$.
In your attempt, since you put $n\in\mathbb N$ inside the set, then $A$ become infinite and is just equal to $\mathbb N\times\mathbb N^*$. I'm not sure this is what you expected.