What difference does "for every $\epsilon$" vs. "for any $\epsilon$" make for the definition of convergence? The definition in my advanced calculus textbook of convergence for sequences is:

A sequence $\{a_n\}$ is said to converge to the number $a$ provided that for every positive number $\epsilon$ there is an index $N$ such that
$$|a_n - a| < \epsilon$$ for all indices $n \geq N$.

Say we replace "for every positive number $\epsilon$" with "for any positive number $\epsilon$". I'm wondering: what difference would that make on the definition of convergence?
 A: I believe your question to be a language one, hence there is no difference in the formulation.
A: These two phrases are the same. They are all "universal quantification".
See also the first item of this list.
Relevant: Difference between "for any" and "for all"?
A: $\epsilon$ belongs to positive real number (strictly greater than 0) so there isn't any difference to it. Its just a matter of wording.
A: They are same. For each, for every and for all mean the same. They are similar phrases used in mathematics. Don't get confused. For any € means it's true for all values of € under the sky.
A: Well, .....
If for every positive $\epsilon$ this is true, then for any positive $\epsilon$ we look at will be true because this is true for every possible $\epsilon$
And if for any positive $\epsilon$ this is true, then the only way that can be true is if it true for every positive $\epsilon$; if there were any positive $\epsilon$ that it wasn't true for, than that $\epsilon$ would be an exception to it being true for any positive $\epsilon$.
......
So even though in english "any one you pick" and "every one you pick" (might) have conceptual contextual differences in a literal linguistic sense, in the world of mathematics (and even the world of practical english applications) they have no practical difference.
SO, they are the same.
....
I pity any non-native english speaker who tries to comprehend the above.
