Quantifier notation: $\forall n \implies \cdot$ versus $\forall n, \cdot$

Question

I'm not sure which of the following two notations is the correct one (or, are both correct?). I've seen both being used by different professors.

1. $\forall \varepsilon > 0\ \exists \bar n \colon \forall n,m \geq \bar n \implies \|x_n - x_m\| < \varepsilon$
2. $\forall \varepsilon > 0\ \exists \bar n \colon \forall n,m \geq \bar n,\ \|x_n - x_m\| < \varepsilon$

If I read the first one out loud it sounds kind of weird: "for all ... implies"?!

Answer 1

They are both correct. I suppose your doubts come from a misinterpretation of the first form indeed it should be spelled in the following way (in natural language)

for all $\epsilon > 0$ exists an $\bar n \in \mathbb N$ such that for every natural numbers $n$ and $m$ *if $n,m \geq \bar n$ then $\|x_n - x_m\| < \epsilon$ or more formally $$\forall \epsilon > 0 \exists \bar n \forall n,m (n,m \geq \bar n \Rightarrow \|x_n - x_m\| < \epsilon)$$

The other form differs in the fact that it hides the implication because it uses a limited quantification, that is a quantification where you impose some restriction.

They are equivalent because generally every formula of the form $$\forall x \in A\ \varphi(x)$$ is actually a short hand for $$\forall x (x \in A \rightarrow \varphi(x))$$.

Answer 2

Both are perfectly intelligible. I'd argue that the first one is slightly better. That might be clearer after saying it correctly in English (after changing a variable name): "for all positive epsilon, there exists a natural number capital N, such that for every natural number m and n, if m and n are greater than or equal to N, then ..." That is, the antecedent of the $\Rightarrow$ is not the quantifiers but rather "$m,n \geq N$".

By contrast, in the second one, the $m,n \geq N$ is absorbed into the quantifiers, which can be a bit confusing sometimes.