I am trying to understand the multiple variable definition of an asymtotic notation. Particularly the definition in wikipedia. It’s also discussed in Asymptotic Analysis for two variables? but I think the answer is wrong. At least it is just corrected in the comments and and referenced to a lengthy answer. What I look for is just the answer for my confusion of the example given here. Wikipedia says,

Big $O$ (and little o, $Omega$, etc.) can also be used with multiple

variables. To define big $O$ formally for multiple variables, suppose

$f$ and $g$ are two functions defined on some subset of

$mathbb{R}^{n}$.

We say $f(vec{x})$ is $O(g(vec{x}))$ as

$vec{x} rightarrow infty$ if and only if $exists M exists C>0$ such that for all $vec{x}$ with $x_{i} geq M$ $textbf{for some i}$ $|f(vec{x})| leq C|g(vec{x})|$.

… For example, if $f(n, m)=1$ and $g(n, m)=n$,

then $f(n, m)=O(g(n, m))$ if we restrict $f$ and $g$ to $(1,

> infty)^{2},$ but not if they are defined on $(0, infty)^{2} .$ This

is not the only generalization of big o to multivariate functions, and

in practice, there is some inconsistency in the choice of defintition.

What I don’t understand is, if we only look **for some i**, why can not we use the domain $(0, infty)^{2} $? For example, if I only take n variable the infinity (i is 0 in this case), then shouldn’t it be fine and $f(n,m) in O(g(n,m))$. Shouldn’t be the definition **not for some i but for all i** then? Do I understand the notion of “for some” in the wrong way?