In the present post you wrote “We also have $ latex \mathbb{N} $ for the set of natural numbers (or positive integers)”. This would have seemed to me to conflict with what I had assumed to be the usual terminology on natural numbers, positive integers and nonnegative integers. Or does it?

Further, your attempted translation of the statement about convergence of a sequence, “However close you want $ latex a_n $ to be to $ latex a $ eventually it will always be that close”, gave me a somewhat misleading impression that one could have intended to mean a given precise distance, in other words, it gave me the feeling that the expression “that close” could be translated back in such a way that the strict order in the corresponding mathematical statement would be replaced by an equality (which is surely not intended). Maybe writing “be within that distance” instead of “be that close” would do more justice to the original mathematical statement?

I decided to go down the implication route in order to correct the other mistake. It’s slightly confusing for the reason you say, but I think it’s reasonably close to how we actually think about that version of the definition. For instance, in a metric space I think we would probably quantify over open balls. And sometimes we sloppily write in a situation like this.

That said, I’d like to point out two things.

First, you have an error in your example “Whenever a,b are real numbers with a less than b…” — you forgot to translate “with” (the precondition that a be less than b). Probably because it is a bit hard to translate into a symbolism you use, when you can either quantify same way over many variables, or quantify with conditions, but usually not both. You can use implication, which will probably confuse things because then you don’t have prenex normal form, you can say “r is between a and b” in a more precise way that doesn’t presume a being less than b, or you can split variables:
“for every a in reals, for every b greater than a, there is…”. But you can’t just leave it as it is.

Second, I strongly recommend you not to use “s.t.” in symbolic sentences. It goes counter to your (excellent) advice not to intermix symbols and words in weak chunks, and trying to defend it as a symbol is very wrong. See 10th comment, point (ii), by David Ullrich, in http://www.mathkb.com/Uwe/Forum.aspx/math/39701/Symbol-for-such-that. In short, just as reverse A doesn’t mean “every” but “for every”, reverse E doesn’t mean “exists” but “exists … such that”. Yes, it’s split, but so is “if … then”.

I know I should be very careful when correcting a Fields medallist about math style, and I know there is a fair bit of tradition in such matters, but I believe this is not only a stylistic question, but also a mathematical one — the same way you feel about “implies” vs. “if … then”. And the tradition is wrong (mathematically).

Still, this is not a major concern in most of mathematics, as one is usually willing to use the framework of set theory to model such sentences. (There are subtle model-theoretic differences between the set-theoretic formalisation and the second-order logic formalisation, but this is an issue that is primarily of concern to logicians rather than to most mathematicians.)

Everyone loves someone (the order of quantifiers has serious implications)

All drinks must be ordered at the bar (you can run up a large bar tab if you are not careful with this one)