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
$$|a_n - a| &lt; \epsilon$$ for all indices $n \geq N$.

Say we replace &quot;for every positive number $\epsilon$&quot; with &quot;for any positive number $\epsilon$&quot;. I'm wondering: what difference would that make on the definition of convergence?
I believe your question to be a language one, hence there is no difference in the formulation.
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 &quot;any one you pick&quot; and &quot;every one you pick&quot; (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.
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I pity any non-native english speaker who tries to comprehend the above.