Different definitions of limit points? I've been doing some self-reading while in holiday over here and going over Stephen Abbot's Introduction to Analysis, in which he gave the following definition for limit points:

Definition 3.2.4. A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighborhood
  $V_\epsilon (x)$ of $x$ intersects the set $A$ in some point other than $x$.

But it was different then what I remember from class so I looked at my professor's notes, and he defined a limit point as:

Let $E\subseteq R$. A point $x \in R$ is called a limit point of $E$ if $\exists y_n \in E (n ≥ 1)$ such that $y_n \to x$.

Using this definition, each point of a set is a limit point, since $\forall a\in A$ you can take $y_n=a \quad \forall n $
Then he defined a closure as just the set of limit points instead of the union as Stephen Abbot did.
Is there a practical difference between the definitions, or is it just a different way of doing things? Is one definition "correct"?
 A: To begin there is a difference between the two definitions as stated. For example, consider the simple set $E = \{ 0 \}$. According to this first definition, $E$ has no limit points. (If $x \neq 0$, then $V_\varepsilon (x)$ is disjoint from $E$ whenever $\varepsilon \leq |x|$; no $V_\varepsilon (0)$ contains a point of $E$ different than $0$.) However the constant sequence $y_n = 0$ witnesses that $0$ is a limit point of $E$ according to the second definition. But this is a result of allowing a point to witness itself being a limit point of a set.
Below I'll use the slight modification of the first definition:

A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighborhood
  $V_\epsilon (x)$ of $x$ intersects the set $A$.

(As phrased in the question, the first definition yields what are sometimes also called accumulation points. The sequential definition of these would additionally require that $y_n \neq x$ for all $n$. To my knowledge, in different texts the the term "limit point" may be used exclusively for these "accumulation points", or for the more general notion.)

Modulo the changes I've mentioned above, these two definitions of limit point are equivalent in the context of metric spaces (or, more generally, in the context of Fréchet–Urysohn spaces).
Let $(X , d)$ be a metric space, and let $E \subseteq X$.


*

*Suppose that $x$ is a limit point of $E$ in the sense of the first definition. Then for each $n$ we have that $V_{1/n} ( x ) \cap E$ is nonempty, so pick some $y_n$ in this intersection. Then $( y_n )_n$ is a sequence of points of $E$, and $y_n \rightarrow x$. Meaning that $x$ is a limit point of $E$ in the sense of the second definition.

*Now suppose that $x$ is a limit point of $E$ in the sense of the second definition. Then there is a sequence $( y_n )_n$ of points of $E$ such that $y_n \rightarrow x$. Given $\varepsilon > 0$ there is an $N$ such that $d ( x , y_n ) < \varepsilon$ for all $n \geq N$. In particular $y_N \in V_\varepsilon (x) \cap E$. This implies that $x$ is a limit point of $E$ in the sense of the first definition.



In the broader class of topological spaces, something closer to the first definition is used. The problem here being that there is generally not a useful notion of $\epsilon$-ball about a point $x \in X$. Instead, we say that $x$ is a limit point of $E$ is every open neighborhood of $x$ has nonempty intersection with $E$. Similarly, the notion of convergence of sequences has to be slightly altered to instead say

$x$ is a limit of the sequence $(y_n)_n$ if for each open neighborhood $V$ of $x$ there is an $N$ such that $y_n \in V$ for all $n \geq N$.

In general, if $x$ is a limit point of $E$ according to the second definition, then it is always a limit point of $E$ according to this rephrased first definition. But the converse need not hold. For example one can consider the ordinal space $\omega_1+1 = [ 0 , \omega_1 ]$ consisting of all countable ordinals together with the least uncountable ording. Here $\omega_1$ is a limit point of $E = [ 0 , \omega_1 )$ according to the first definition, however there is no sequence in $E$ which converges to $\omega_1$. (See also the Wikipedia article on the First uncountable ordinal.)
So, the first definition is the better definition as far as general topological spaces is concerned, but if you're only concerned with metric spaces (or the real line $\mathbb R$), they are the same, and context will determine which is more useful. (E.g., sometimes it might be easier to contruct a sequence which obviously converges to the desired point.)
(As a final aside, there is a generalized notion of convergence in topological spaces by nets which completely captures the notion of limit point. This is sometimes referred to as Moore–Smith convergence.)
