Can I define the limit of a sequence like this? 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.


*

*Define $\mathbb{R}$ by using the least-upper-bound (LUB) property.

*Prove Cantor's intersection theorem (CIT) by using LUB.

*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.

*Prove Bolzano-Weierstrass theorem (BWT) for infinite sets by using CIT.

*Definition: a sequence is just a map $f:\mathbb{N}\to\mathbb{R}$. Note that $f$ and $f(\mathbb{N})$ are different.

*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.

*Prove BWT for sequences: if $f(\mathbb{N})$ is finite, the pigeonhole principle; if not, the BWT for infinite set.

*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.

*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 $\epsilon$.

 A: 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.
A: 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.
