I am having difficulty understanding the definition of convergence. I've been rereading and looking at examples during this past week and I haven't made any progress.

**Definition:** We say that {${a\_{n}}$} converges to a point $a \in \mathbb{R}$ if for any $\epsilon$, there exists a positive integer $N$ such that for any $n \in \mathbb{N}$ with $n\geq N$, one has $|a\_{n}-a|< \epsilon$.

My questions:

(1) What role does $\epsilon$ play? Is that the actual limit? I thought that $a$ is what we are "assuming" is the limit?

(2) Why does $n \geq N$? I'm asking this because we were just given the definition of what it means to be convergent (in Real Analysis, not the Calculus sequence), no formal proof.

(3) In various examples they are trying to set $N$ to be less than or equal to $\epsilon$. Why?

My main issue is that I don't understand how the components of this definition work. I can follow the examples, but I'd rather understand why it works then just take it on blind faith.

Thank you for any input/suggestions.