Now, by this definition, $\displaystyle |\{a_n\}-L|<\epsilon$. Now, epsilon is just some small number that represents how far away the sequence is from the limit. When we find a value for $\displaystyle N$ we have the largest possible value for which the sequence is less than epsilon. For every $\displaystyle n>N, n\in \mathbb{N}$ the sequence will be less than epsilon and so we can write $\displaystyle -\epsilon +L<\{a_n\}<\epsilon +L$.

What would you guys add or change in my understanding? Thanks!