# Limit of a sequence (definition)

Let $$(a_n)_{n \in \mathbb{N}}$$ be a sequence.

According to the definition of limit, it is said that $$\lim_{n \rightarrow \infty} a_n=L \Leftrightarrow \left(\forall \varepsilon>0, \ \exists n_0 \in \mathbb{N}, \ n \ge n_0 \Rightarrow \left|a_n-L \right|<\varepsilon \right).$$

About the details, does it really matter, whether you say $$n \ge n_0 \ \mathrm{or} \ n > n_0 ?$$

And why would $$n_0$$ have to be an integer?

## 2 Answers

We don't need that $$n_0$$ is an integer and it doesn't matter whether we use $$\ge$$ or $$>$$ both lead to an equivalent definition.

For the first part of your question, no it does not particularly matter as both statements will hold true for some epsilon greater than 0.

$$n_o$$ refers to the $$n_o$$ term of the sequence and thus is an integer. The limit $$L$$ however can be a real number.

• Not necessarly $n_0$ needs to be an integer, the definition works fine alfo with $n_0 \in \mathbb R$. But of course we can also use $n_0 \in \mathbb N$ as in te given example.
– user
Sep 8, 2020 at 13:50