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Consider the definition of the limit of a sequence $(x_n)$ in real analysis: $$\lim x_n=x\iff\forall \epsilon>0\exists N:\forall n\geq N: |x_n-x|<\epsilon.$$ Now, I am quite sure I get the idea of the limit, how to write it in words, how to do questions in it, etc: this is not a question about real analysis. What I am not sure about is my statement of the definition in this kind of logical notation. In which order the quantifiers $\forall$ and $\exists$ should be placed. When should and shouldn't I place $:$ after something? Does it matter if I place $\forall\epsilon>0$ after $\exists N$? Does it matter if I place $\forall n>N$ after $|x_n-x|<\epsilon$?

I realise there are many similar questions previously posed on this site, like here and here. However these aren't duplicates of my question, as I don't think their answers address my doubts very well. I am just looking for a quick synopsis about the basic thought process I should go through when trying to formulate a statement like above (again, that is just an example), and the basic ideas why (or why not) the order of certain placements matter. In addition, if the order doesn't matter, are there any commonly agreed upon "style guides" as to how one should write them to be more "aesthetically" pleasing? Many thanks in advance.

YiFan Tey
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  • Yes, order of quantifiers does matter! – Angina Seng Nov 04 '18 at 10:17
  • Thank you, could you please briefly explain why? – YiFan Tey Nov 04 '18 at 10:19
  • "for a random distance from $x$ the sequence must eventually be close enough to $x$". So, for every $\varepsilon > 0$ (random distance) we don't require that all of the sequence lies within the distance, but that it eventually does, so there exists some index $N$ st so and so forth.. – AlvinL Nov 04 '18 at 10:21
  • @AlvinLepik Yes, I completely understand the real analysis part of it, and the definition in English. I'm not sure about whether the way I wrote it in terms of logical symbols as above correctly translate to the English meaning, however. What I am looking for is a basic guide on how I should interpret (and form) these statements, especially the confusing part with the quantifiers. – YiFan Tey Nov 04 '18 at 10:23

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I like to see $\forall \varepsilon>0: \exists N: \forall n \ge N: |x_n -x| < \varepsilon$ as a description of a game of sorts: if you want to show the limit is $x$ you get a challenge from your opponent, in the form of some $\varepsilon>0$. For that $\varepsilon>0$ you (the player) has to find somehow an $N$ such that whatever move $n$ your opponent plays (you only know that $n \ge N$ for the $N$ you just gave), you can show that $|x_n -x| < \varepsilon$ and you "win" that game.

The statement $\lim_n x_n = x$ as thus defined is true when the player can win this game against any opponent, and he loses (the statement is false) when the opponent has a winning strategy: he can find some $\varepsilon >0$ such that whatever $N$ the player chooses, he can find some larger $n$ where $|x_n -x| < \varepsilon$ fails.

You can see that the choice of $N$ (the player's move) is only done after the opponent chooses the $\varepsilon$, and the $n$ he then has to deal with depends on this $N$ in turn. So the order is crucial, especially in such $\forall \exists$ alternations. The order of quantifiers corresponds to the order of "moves" in the game.

Such game interpretations to determine truth are actually used in some logics (that allow for infinitely many quantifiers in a formula, e.g.), but it can help think about such finite statements as well, I find.

Henno Brandsma
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My algebra professor taught me this style $$(\forall \varepsilon >0)(\exists N\in\mathbb N)(\forall n\in\mathbb N) (n >N \implies P(n,\varepsilon)) $$ If I were to use the $:$ symbol as a stand-in for "such that.." I would do the following. $$\forall\varepsilon >0\ \exists N\in\mathbb N\ : \forall n\in\mathbb N (n > N \implies P(n,\varepsilon)) $$ This is really a matter of taste, I guess.

Order of quantification does matter. If we were to write (I'm now using the notation I'm used to) $$(\exists N\in\mathbb N)(\forall \varepsilon >0)(\forall n\in\mathbb N)(n > N \implies P(n,\varepsilon)) $$ This is completely different. Now we ask for an index $N$ independent of $\varepsilon$, whereas before we picked $N$ according to our choice of $\varepsilon$.

AlvinL
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