Intuitive understanding of Limit of sequence 
Statement :- the number $\alpha$ is called limit of a sequence ${x_1, x_2, \dots, x_n, \dots}$ as $n \to \infty$, $\alpha = \lim_{n \to \infty} x_n$,  if $\forall \epsilon \gt 0, \exists N_\epsilon $ and the inequality $|x_n - \alpha| \lt \epsilon$ is true $\forall n \gt N_\epsilon$

This statement is giving me nightmares $\ddot \frown$ . I know what those symbols mean , but i can neither understand why do we take that specific inequality to prove the limit of a sequence ? nor the condition, $\forall n \gt N_\epsilon$ ? 
It would be nice if somebody can help me understand this statement intuitively. $\ddot \smile$. Thanks for the help    
 A: Quick note: I have never seen $\lfloor \epsilon \rfloor$ used in this way and i suspect there is a mistake. The expression $\lfloor \epsilon \rfloor$ usually means the floor function of $\epsilon$ and there is no reason to believe $N$ in the usual definition should be this value.
However, the statement for a sequence approaching a limit $\alpha$ is that 
$$
\forall \epsilon>0
$$
or for any $\epsilon$ arbitrarily small but larger than zero, we can find
some $N$ generally depending on this $\epsilon$ for which you can guarantee "closeness" to the limit for all terms in the sequence with indices larger than that $N$, or mathematically
$$
n\geq N\Rightarrow |a_n-\alpha|<\epsilon
$$
A: $\forall \epsilon> 0, \exists N>0: n>N \implies |x_n - a|<\epsilon$
For any (small) $\epsilon$, we can find some $N,$ (possibly large) such that every number in the sequence past $N$ is within our small radius of the target.
I say $x_n$ is converging to $a$
Why? because far enough in the tail, everything is close to $a.$  
How close?  You tell me.  Make $\epsilon$ as small as you want it to be, and I will tell you how far out the tail you need to start paying attention.
A: I try to explain this definition as I understood it in the first semester of my studies:
$|x_n - \alpha|$ is the distance between $x_n$ to the limit $\alpha$. By our intuitive understanding of convergence we expect, that this distance becomes smaller and smaller the larger the $n$ is. And this is what the definition says!
You can choose any "maximal distance" $\epsilon > 0$ arbitrary, and what you get is an $N \in \mathbb{N}$, so that the distance betweeen $x_n$ and the limit $\alpha$ is smaller than your $\epsilon$, but only for the $x_n$'s with $n \ge N$.
A: See my image, essentially, you are saying that at some point N, the distance between your convergent value a, and the value of the sequence X sub n, (ie |Xn - a|), will ALWAYS be less than epsilon, in other words, at values of x less than N, there are a FINITE number of terms, but past that point, an infinite number of terms in the sequence exist, all of which are within a certain distance, epsilon, to your convergent value a.
