precise vs ordinary (?) definition of the limit

Question

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 $\lim_{n \to\infty}$?

Answer 1

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.

Answer 2

Let's start from scratch. Here is a "precise" definition of limit: Let f be a real-valued function whose domain is a subset of the real numbers and let a and L be real numbers. We say the limit of f at a is L provided that for each open interval containing L there is an open interval containing a with the property that if w is any number in this interval other than a itself, w is in the domain of f and f(w) is in the the open interval containing L. There are three things to notice: in order for this definition to make sense the words "the limit" require that we make sure that f cannot have two different limits at a; there is no mention of the x-axis in the definition ;also there is no x --> a, neither x nor anything else approaches a. Nothing moves. Finally we note that the definition says absolutely nothing about a. We are given that it is a real number, but that's all we know. After a while we notice the behavior of functions like f(x)= 2+ 1/x and we would like to say that the limit of f at infinity is 2. To make the definition above work, we take anything you like other than a real number and call it infinity. (I had an algebra teacher who used a little red rooster and that worked fine. You can't go wrong, so long as you don't think infinity is a number.) For each real number r we say that the set of all real numbers greater than r, together with the red rooster, is an open interval about the red rooster. With this convention, the definition of limit works as we want it to. A sequence is just a function whose domain is the set of nonnegative integers in some open interval about the red rooster, so the definition applies to sequences as a special case.