Question about quantifiers' order in a sentence I'm confused about the order in which we are allowed to write the quantifiers in sentences. During my first analysis lecture, our teacher told us we should never write quantifiers on both sides of an expression, but during my second lecture of analysis, it seems like our new teacher didn't really care about that. For example, he wrote things like:

A sequence of functions $\{f_n\}$, $f_n: D \rightarrow \mathbb{R}$, converges pointwise to $f: D \rightarrow \mathbb{R}$ if $\forall x \in D, \forall \epsilon > 0, \exists n_0 \in \mathbb{N}$ such that $|f_n(x)-f(x)| < \epsilon$ $\forall n \ge n_0$

or

A sequence of functions $\{f_n\}$, $f_n: D \rightarrow \mathbb{R}$, converges uniformly to $f: D \rightarrow \mathbb{R}$ if $\forall \epsilon > 0, \exists n_0 \in \mathbb{N}$ such that $|f_n(x)-f(x)| < \epsilon$ $\forall n \ge n_0, \forall x \in D$

Since it's already difficult for me to understand the concepts of pointwise/uniform convergence and what is the difference between the pointwise and uniform convergences of sequences of functions, I don't think that reversing some quantifiers in the definitions is going to help. (It seems like Wikipedia would put them in a different order.) However, I don't really understand either why we shouldn't write quanitifers on both sides of an expression... Could someone explain me the logic behind this? Thanks (a lot) in advance!
 A: I'd say, this is not a question of logic, but of writing style in general. 
For a "sentence" as used in logics --a formula without free variables-- the usual syntax definitions are unambiguous and it's clear that writing the quantifier at another position is just wrong. 
But that's not the point here. You are not writing a logical formula, but a sentence in the so-called object language and you use mathematical symbols like \forall just as an abbreviation for the natural language expression "for all". 
Typically, students new to logics tend to mix up these two levels syntax and semantics and we are very strict at the beginning to distinguish these by allowing quantifier symbols only at the syntax level until we can make sure that this distinction is clear to everybody.
Mathematicians, in many instances, however do not care about this distinction that much.
To answer your question more concretely, I'd also say that your samples are somewhat hard to read because of the quantifiers on both sides. Another style rule is that you should avoid putting mathematical different mathematical expressions next to each other within text in between like "$|f_n(x)−f(x)||f_n(x)-f(x)| < \epsilon$ $\forall n \geq n_0$". I'd thus rather write

A sequence of functions $\{f_n\}, f_n: D \rightarrow \mathbb{R}$, converges pointwise to $f: D \rightarrow \mathbb{R}$ if $\forall x \in D, \forall \epsilon > 0, \exists n_0 \in \mathbb{N}, \forall n \ge n_0$ such that $|f_n(x)−f(x)||f_n(x)-f(x)| < \epsilon$ 

or 

A sequence of functions $\{f_n\}, f_n: D \rightarrow \mathbb{R}$, converges pointwise to $f: D \rightarrow \mathbb{R}$ if $\forall x \in D, \forall \epsilon > 0, \exists n_0 \in \mathbb{N}$ such that $|f_n(x)−f(x)||f_n(x)-f(x)| < \epsilon$ for all $n \ge n_0$

