# precise vs ordinary (?) definition of the limit

I am very confused about this "precise definition" (the $\epsilon$-$\delta$-definition) of the limit thing. So far I have been dealing with sequences and the definition of a limit I was presented so far was:

$\lim_{n \to \infty} (a_n)_{n \in \mathbb{N}} = L \iff \forall \epsilon > 0. \exists N \in \mathbb{N}. \forall n \geq N. |a_n - L| < \epsilon$

This obviously does not cover indices that are not $\infty$. But is that all the difference to the "precise definition" or is there more to it? I looked for proofs of the limit laws, like "the limit of the sums is the sums of the limits" and all proofs I found used the "precise definition". Is this only to be more general and cover non infinity indices which are being approached ($\lim_{n \to c \neq \infty}$) or is it necessary to use the precise definition even for the case of $\lim_{n \to\infty}$?

• unclear what you are asking – GEdgar Dec 3 '16 at 18:19
• @GEdgar I am asking 1) Is the epsilon-delta definition of the limit' required to proof for instance the limit laws such as the one mentioned if one is only concerned with infinity as the index one approaches. 2) is the only difference between the epsilon-delta definition and the one I wrote down the fact that the one I wrote done does not cover non-infinity indices as the index that is being approached. – user3578468 Dec 3 '16 at 18:22

OK, the "$\epsilon$-$\delta$ definition" is for cases where the $\delta$ condition makes sense: $|x-a| < \delta$. This is meaningless for $a=\infty$. So a slight variant is used. You could call it the "$\epsilon$-$N$ definition" or something. Which one to use is determined by which one makes sense. We can imagine a third, similar, definition for $$\lim_{n\to\infty} a_n = \infty .$$ Then neither the $\delta$ condition nor the $\epsilon$ condition makes sense. See if you can write the definition that will apply in that case.