Can I define the limit of a sequence like this?

Question

It is well-known that a sequence has a limit if and only if it is bounded and has a unique limit point. I think this is a better definition of the limit of a sequnece, comparing with the $\epsilon-N$ one.

When teaching mathematical analysis, we can prove the Bolzano-Weierstrass theorem first, which is intuitively trivial. The B-W theorem asserts that a bounded sequence has at least one limit point, then, if a sequence has a unique limit point, it deserve a better name, and we name it a converget sequence.

I have checked all my analysis textbooks, and nobody defines limit like this? Why? Is there anything wrong with this approach? Thank you!

EDIT: Let me give more details.

1. Define $\mathbb{R}$ by using the least-upper-bound (LUB) property.
2. Prove Cantor's intersection theorem (CIT) by using LUB.
3. Definition: a point $x$ is called a limit point of a set $A$ if for any $\epsilon>0$ the set $\{y\in A \mid |x-y|<\epsilon\}$ is an infinite set. You can prove that a point of $A$ is either limit or isolated.
4. Prove Bolzano-Weierstrass theorem (BWT) for infinite sets by using CIT.
5. Definition: a sequence is just a map $f:\mathbb{N}\to\mathbb{R}$. Note that $f$ and $f(\mathbb{N})$ are different.
6. Definition: a point $x$ is called a limit point of a sequence $f$ if for any $\epsilon>0$ the set $\{n\in\mathbb{N}\mid |x-f(n)|<\epsilon\}$ is infinite. A limit point of $f$ is either a limit point of $f(\mathbb{N})$ or an isolated point of $f(\mathbb{N})$ which is hit by $f$ infinite times.
7. Prove BWT for sequences: if $f(\mathbb{N})$ is finite, the pigeonhole principle; if not, the BWT for infinite set.
8. Definition: a sequence $f$ is convergent if it is bounded and has a unique limit point. This limit point, say $x$, is called the limit of this sequence.
9. Theorem: a sequence $f$ is convergent if and only if for any $\epsilon>0$ the set $\{n\in\mathbb{N}\mid |x-f(n)|\ge \epsilon\}$ is finite. Proof: Suppose $f(\mathbb{N})\subset [a,b]$. For a point $y\in[a,x-\epsilon]\cup[x+\epsilon,b]$, it must be an isolated point of $f(\mathbb{N})$ and is hit by $f$ finite times. According to BWT, $[a,x-\epsilon]\cup[x+\epsilon,b]$ cannot contain infinite isolated points of $f(\mathbb{N})$. Then you can find the $N$ for $\epsilon$.

Answer 1

In real analysis, or indeed in any metric space, the $\epsilon, N(\epsilon)$ definition has the attractive feature that it is fairly intuitive to anybody familiar with the $\delta(\epsilon)$ definition of derivative, which for practical purposes is everybody taking the course. It is also easier to apply than the unique limit point definition, in that if you give a homework or exam problem involving proving a limit, most of the time the student would need to do something ling the $\epsilon, N(\epsilon)$ definition to say something about the limit points first, anyway.

The big advantage of the unique limit point is that it is applicable even for a sequence in a topological space that is not a metric space. And I would expect that many topology texts (for example, a definition in one of the problems in Munkries) would in fact use your suggested definition, particularly if the author sees fit to discuss sequences before introducing the notion of metrics.

Answer 2

Here is the problem with your definition:

Consider $$\lim_{n \to \infty} {1\over n}$$ Using your idea, we could use induction to show that the the sequence is bounded.

Challenge #1: Prove that to me that the sequence has a unique limit point. The best definition I am aware of (topological) can be shown to be equivalent to the sequential definiton.

Challenge #2: Prove to me, using your definition, that $$\lim_{n \to \infty} {1\over n} = 0$$ That is the benefit of, what you call $N-\epsilon$, definition of a limit. It proposes and proves the limit of the sequence using the Archimedean Principle:

Given any $\epsilon > 0$ by the Archimedean Principle there exists $N > 1/\epsilon$, where for all $n \geq N$, $${1 \over n} - 0 < \epsilon$$

As shown, our normal definition both verifies that a proposed limit is true, rather than just asserting its existence.