TITLE: "such that" in logical statements
QUESTION [5 upvotes]: How exactly do I put this into logical statements? 
For any x, there exist an n such that P(x). 
I know to start with ∀x ∃n, but where do I go from here?

REPLY [2 votes]: How exactly do I put this into logical statements? 
For any x, there exist an n such that P(x). 
I know to start with ∀x ∃n, but where do I go from here? 

You add $P(x)$
$$\forall x\exists n\;P(x)$$
A space typeset between the n and P improves legibility.   Sometimes this may not seem enough, so you might write $\forall x\exists n{:}P(x)$ or $\forall x\exists n{.}P(x)$ or use some other punctuation mark to clearly distinguish between the quantified variables and the predicate bound by them (I prefer the colon).   These symbols are optional; they're just to add clarrity.   Parenthesis do the same work.
$$\forall x\exists n\,[P(x)]$$