As an example, take a look at the Cauchy's convergence test:
$\forall_{\varepsilon>0} \exists_{n_0\in\mathbb{N}} \forall_{m,n \geq n_0} \colon \left|a_m-a_n \right|<\varepsilon $
or
$\forall_{\varepsilon>0} \colon \exists_{n_0\in\mathbb{N}} \colon \forall_{m,n \geq n_0} \colon \left|a_m-a_n \right|<\varepsilon $
or
$\forall_{\varepsilon>0} \exists_{n_0\in\mathbb{N}} \forall_{m,n \geq n_0} \left|a_m-a_n \right|<\varepsilon $
Is there a difference in the meaning of these three statements? (If yes, please explain the difference)
When should I make a colon in a mathematical statement and when not?
(I've found the first and the second in Wikipedia)