TITLE: What can $\delta$ depend on and not on in $\delta-\epsilon$ formulation of limit?
QUESTION [6 upvotes]: in the  delta-epsilon formulation for $\lim_{n\to \infty} x_n = L$,

$\forall \epsilon >0, \exists n_0 \in \mathbb N,$
$$ |x_n-L| \leq \epsilon, \ \forall n > n_0$$

$n_0$ depends on $\epsilon$.
What else can it depend on?
Can $n_0$ depend on $x_n$?
What  can it not depend on?
Thanks.

REPLY [4 votes]: In a statement of the type $\forall \epsilon\exists \delta\forall x$, the $\delta$ may depend on anything that is written before it (as well as things you make up while defining $\delta$), but not anything written later (i.e. $\delta$ may not depend on $x$). Therefore, it is much simpler to think of the things that $\delta$ may not depend on (the things that you define after you define $\delta$). Strictly speaking, you may want to write something like
$\forall X$ metric spaces, $\forall x\in X$, $\forall (x_n)$ sequences in $X$ converging to $x$ $\forall \epsilon$ positive real numbers $\exists N\in\mathbb N$ such that $n\geq N$ implies $d(x,x_n)<\epsilon$.
When written like this, $N=N_{X,x,(x_n),\epsilon}$ may depend on all of the four previously named objects. But thinking about $N$ in this way quickly gets tedious. Note also how $x$ 'depends' on $X$, $(x_n)$ 'depends' on both $X$ and $x$, and that $n$ is not a named object at the point where $N$ is named (it is just a part of common notation for sequences). Note also that we could have included even further objects if we had wanted (say we could have considered different metrics or topologies, or different set-theoretic axioms).
In conclusion, whenever you name a new object, that object may depend on any object that you have already named. If it is important that some object does not depend on any specific previously named object, you should mention it explicitly.
Example: Consider the difference between the following three situations:
1) $\forall \epsilon>0,\exists N_{\epsilon}\in \mathbb N,\forall n\geq N_{\epsilon}: |1/n|<\epsilon$. Namely pick $N_\epsilon=\lceil 2/\epsilon\rceil$.
2) $\forall a>0,\forall \epsilon>0,\exists N_{\epsilon}\in \mathbb N,\forall n\geq N: |1/n|<\epsilon$. Namely pick $N_\epsilon=\lceil \max(a,2/\epsilon)\rceil$.
3) $\forall \epsilon>0,\exists N_{\epsilon}\in \mathbb N,\forall n\geq N: |1/n|<\epsilon$. Namely pick $N_\epsilon=\lceil \max(a,2/\epsilon)\rceil$, where $a>0$.
The only thing which matters in all three cases is that $N$ may NOT depend on $n$, because $n$ is named later than $N$.