Understanding the limit definition of a sequence

Question

Let $(a_n)$ be a sequence and $L$ be a real number. We say $\lim(a_n)=L$ if

$\forall\epsilon>0\exists N\in\mathbb{N} (n\geq N \Rightarrow |a_n-L|<\epsilon)$

I understand that n is the index of the sequence and L is the limit of the sequence and $\epsilon$ is the arbitrary / infinity small value +/- that we use to get close to L (?) but what is N?

If somebody could walk me though the definition so I could understand this better, I would appreciate it.

Answer 1

In words the definition says that for every $\epsilon>0$ you can find a natural number $N$ large enough such that $|a_n -L| <\epsilon \ \forall\ n \geq N$. This means that after a finite number of terms, the sequence gets very close to $L$.

For example, let $a_n=\frac1n$ and $\epsilon >0$ be given. Choose $N \in \mathbb{N}$ such that $N > \frac1\epsilon$.

Then $|a_n-0|=\frac1n \leq\frac1N< \epsilon \ \forall \ n\geq N.$

Answer 2

The idea is that when $n$ gets larger and larger, $|a_n - L|$ gets smaller and smaller. "Small" is captured by $\varepsilon$ and "large" is captured by $N$. That is, when $n$ is at least as large as $N$, $|a_n - L|$ is at least as small as $\varepsilon$.

Answer 3

$N$ is a positive integer that depends on $\epsilon$. Let say $\epsilon = 0.01$, and $N_{\epsilon}=10$.

We can conclude that $$|a_{10}-L| < \epsilon,$$

since $10 \geq 10$.

Also,

$$|a_{101}-L| < \epsilon,$$

since $101 \geq 10$

We can say similar thing for $a_{n}$ for any $n$ that is bigger than $N_{\epsilon}$.

The whole idea is as $n$ is large enough, $a_n$ get arbitrarily close to $L$. The $N$ is used to indicate when is $n$ large enough. How big should my $n$ so that distance of $a_n$ and $L$ is less than $\epsilon$.