TITLE: Limit point versus limit and implications for convergence
QUESTION [15 upvotes]: I understand that similar questions to this have been asked here (1) and here (2) but I believe my question is different because with regards to (1) I am not asking about the set $X$ that contains the points of my sequence and with regard to (2) because I have the distinct definitions of limit and limit point in mind. That said:
My Question: My textbook, Fundamental Ideas of Analysis by Michael Reed, defines a limit point as:

Definition: Let $(a_n)$ be a sequence of real numbers. A number $d$ is called a limit point if, given $\epsilon > 0$ and an integer $N$, there exists $\color{red}{\mathrm{ an }} \ n \ge N$ so that $|a_n - d| \le \epsilon$. 

The definition of a limit is as follows:

Definition: We say that a sequence $(a_n)$ converges to a limit $a$ if, for every $\epsilon \ge 0$, there is an integer $N(\epsilon)$ so that $|a_n - a| \le \epsilon \color{red}{\text{  for all  }} n \ge N$. 

From these definitions, and given this question:

If a sequence converges it has exactly one limit point. Is the converse true? If not, provide a counterexample

Can I conclude that the converse is indeed true simply by appealing to the definitions, or do I need a tighter proof. And if I do, I'm a little stuck on how to refine this proof.

REPLY [4 votes]: You could also pick the sequence 
$ (1 + (-1)^n).n $
which doesn't converge but has 0 as limit point.