What difference does "for every $\epsilon$" vs. "for any $\epsilon$" make for the definition of convergence?

Question

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?

None. "for every" and "for any" are english phrases, that for all practical mathematical purposes mean the same thing. I suppose one can argue "for every" implies in english that we have ever case, and "for any" might imply we only have one but whichever one case we have will work, but in practical terms they mean the same thing. If we arbitrarily pick one it will work, but the only way we can know that any we pick will work has to be because every case works... it not every case worked we can't assure that any case, no matter which, would work. — fleablood, Sep 13 '20 at 01:07
You could also phrase this as: "if $\epsilon > 0$ then there exists a positive integer $N$ such that if $n$ is an integer and $n \geq N$ then $|a_n - a | < \epsilon$." — littleO, Sep 13 '20 at 01:19
See Is the universal quantification symbol ∀ known as “for any x” or “for all x” in First Order Logic & why is this different to Discrete Math? and Difference between "for any" and "for all". — Mark S., Sep 13 '20 at 02:22

score 2 · Accepted Answer · answered Sep 13 '20 at 01:02

2

I believe your question to be a language one, hence there is no difference in the formulation.

answered Sep 13 '20 at 01:02

Fer Stein

126

score 0 · Answer 2 · answered Sep 13 '20 at 01:58

0

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"?

answered Sep 13 '20 at 01:58

The second answer to the question you link to makes a good point: "Either do not use "any" in Math or explain to the reader what it means in your context" because it is ambiguous. – Misha Lavrov Sep 13 '20 at 02:24

score 0 · Answer 3 · answered Sep 15 '20 at 08:00

0

$\epsilon$ belongs to positive real number (strictly greater than 0) so there isn't any difference to it. Its just a matter of wording.

answered Sep 15 '20 at 08:00

RiRi

417

score 0 · Answer 4 · answered Sep 15 '20 at 17:47

0

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.

answered Sep 15 '20 at 17:47

vidya

33
5

score -1 · Answer 5 · answered Sep 13 '20 at 01:14

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.

+1. And my elaboration: Difference between For any, For all, For every, For each. — ryang, Feb 26 '23 at 11:34

What difference does "for every $\epsilon$" vs. "for any $\epsilon$" make for the definition of convergence?

5 Answers5