Consider a collection $A=\{A_1,\dotsc,A_k\}\subseteq\mathbb{N}$. Suppose that every element of $A$ satisfies a property, say, they are nonzero. Which of the following notation would you prefer?
I think it does not change anything, but it bothers me a little.
On
Since Quantifiers are used at the beginning of a formula you could consider it as wrong to write something like $A_i \neq 0$ $\forall i = 1, \dots, n$. You could fix that notational problem writing $A_i \neq 0$ for all $i \in \{1, \dots, n\}$.
I neither would use the notation $A_i \neq 0, i= 1, \dots, n$.
Alternatively you could write something like $$\forall i \in \{1, \dots, n \}: A_i \neq 0$$ which most people would consider as more elegant.
On
Two alternatives, which I prefer, are$$A_i\neq0\quad(i=1,...,n),$$$$A_i\neq0\quad\text{for}\; i=1,...,n.$$The issue with your first example is that the comma normally separates statements of equal weight, and functions as “and”, as in$$x=a,\quad y=b,\quad z=c.$$In your second example, the $\forall$ is wrongly spaced: it is too attached to the $i$. Replacing the word for by $\forall$ in a sentence looks cleverer than replacing it by 4, but neither makes reading easier. Of course, the $\forall$ symbol has its uses, particularly in logic.
In my opinion, neither. I would use$$(\forall i\in\{1,2,\ldots,k\}):A_i\neq0.$$For me, $i=1,2,\ldots,k$ doesn't make sense; $i$ can be only one of the numbers $1,2,\ldots,k$.