Q:

The definition of a sequence converging.

The sequence $\{a_n\}$ converges to t he number L if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$,
$$n > N \implies |a_n - L| < \epsilon$$
If no such number $L$ exists, we say that $\{a_n\}$ diverges. If $\{a_n\}$ converges to L, we write $$\lim_{n\to\infty}a_n=L$$
I am having trouble understanding this definition in my textbook. Specifically when they use $\epsilon$, $N$, and $n$. Is $\epsilon$ the values in the sequence, $n$ the n-th term of the sequence, and $N$ an arbitrary integer? Could someone explain this to me? 
*edit
Show that $lim_{n\to\infty}\frac{1}{n}=0$.
The implication will hold if $\dfrac{1}{n}< \epsilon$ or $n > \dfrac{1}{\epsilon}$. If $N$ is any integer greater than $\dfrac{1}{\epsilon}$, the implication will hold for all $n>N$.

A:

Think of it as a game between us:
I choose a small number $\epsilon$ then you have to see if you can find an $N$ such that for all $n > N$ the terms of the sequence are within $\epsilon$ of $L$.
If you can always do so, then the sequence converges - you win.

