What is the difference between 1) given any , 2 ) for every 3) for all in definitions?

Question

I have a lot of confusion.

Definition:

A sequence $\{x_n\}$ of points in a metric space is said to converge if there is a point $p \in S$ with the following property: For every $\epsilon> 0$ there is an integer $N$ such that $d(x_n, p) < \epsilon$ whenever $n \geq N$.

What if we write:

1) Given $\epsilon> 0$, there exists an integer $N$.

2) For all $\epsilon> 0$, there exists an integer $N$.

Would it make any difference?

None. It's usually to avoid repetitions, considered as bad style. — Bernard, Aug 21 '15 at 15:14

score 1 · Answer 1 · answered Aug 21 '15 at 15:20

No, it wouldn't, they are different words but they express the same thing. To convince yourself, prove the equivalences between the 3 propositions (for e.g. 1 => 2 => 3 => 1), but they are trivial.

Read about quantifiers if you want. Those 3 propositions, translated to predicate logic, are equal.

score 1 · Answer 2 · answered Aug 21 '15 at 15:25

1

There is no difference. Mathematically, they are three different ways of denoting the $\forall$ quantifier.

answered Aug 21 '15 at 15:25

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What is the difference between 1) given any , 2 ) for every 3) for all in definitions?

2 Answers2