I've been scouring around looking for a definition for Big-O when you have multiple input variables.
For context, I'm an undergraduate student.
Wikipedia mentions $$f(\mathbf{x})\text{ is }O(g(\mathbf{x}))\quad\text{ as }\mathbf{x}\to\infty$$ if and only if there exist constants $M$ and $C > 0$ such that $|f(\mathbf{x})| \le C |g(\mathbf{x})|$ for all $\mathbf{x}$ with $x_i \geq M$ for some $i$.
Should it be "for some $i$" or "for all $i$". Intuitively, I think it would be the former but I can't find any source for this.
The cited link on Wikipedia mentions pg. 48 of the 3rd edition of CLRS. However, pg. 48 of that book does not mention this definition.
This answer seems like it mentions the "for all $i$" definition: https://cs.stackexchange.com/a/30039.
This answer on a similar post also seems like it argues for "for all $i$". Although a comment on it argues otherwise citing a post which does not support their claim and instead cites a paper (https://people.cis.ksu.edu/~rhowell/asymptotic.pdf) which seems like it proposes a definition with "for all $i$"?
It says "such that for all $n_1 \geq N, ..., n_k \geq N$" which lead me to believe that but perhaps I'm understanding it wrong.
