# What is meant by the limit of an infinite sequence?

The limit of an infinite sequence tells us about the long term behaviour of it.

Given a sequence of real numbers ${a}_{n}$, it's limit ${\lim}_{n \to \infty} {a}_{n} = \lim {a}_{n}$ is defined as the single value the sequence approaches (if it approaches any value) as we make the index $n$ bigger. The limit of a sequence does not always exist. If it does, the sequence is said to be convergent, otherwise it's said to be divergent.

Two simple examples:

• Consider the sequence $\frac{1}{n}$. It's easy to see that it's limit is $0$. In fact, given any positive value close to $0$, we can alway find a great enough value of $n$ such that $\frac{1}{n}$ is less than this given value, wich means that it's limit must be less or equal to zero. Also, every term of the sequence is greater then zero, so it's limit must be greater or equal to zero. Therefore, it is $0$.

• Take the constant sequence $1$. That is, for any given value of $n$, the term ${a}_{n}$ of the sequence is equal to $1$. It's clear that no matter how big we make $n$ the value of the sequence is $1$. So it's limit is $1$.

For a more rigorous definition, let ${a}_{n}$ be a sequence of real numbers (that is, $\forall n \in \mathbb{N} : {a}_{n} \in \mathbb{R}$) and $\epsilon \in \mathbb{R}$. Then the number $a$ is said to be the limit of the sequence ${a}_{n}$ if and only if:

$\forall \epsilon > 0 \exists N \in \mathbb{N} : n > N \implies | {a}_{n} - a | < \epsilon$

This definition is equivalent to the informal definition given above, except that we don't need to impose unicity for the limit (it can be deduced).