---
tag: 09G6KHS6GY
kind: statement
role: proposition
name: Epsilon-N definition of the limit of a sequence
msc: [40A05]
uses: [0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Real numbers
---

## Statement

Let $(a_n)$ be a real sequence and let $L$ be a real number.
Then $(a_n)$ converges to $L$, in the sense of the [[limit of a sequence|0XKWSSY8P2]], if and only if the following holds.
For each real number $\varepsilon > 0$ there is a natural number $N$ such that $|a_n - L| < \varepsilon$ for all $n \ge N$.

## Remarks

The proof uses two facts. Each interval $(L - \varepsilon, L + \varepsilon)$ is open and contains $L$. Each open interval that contains $L$ contains such an interval.
In a metric space $(X, d)$ the same statement holds with $d(a_n, L)$ in place of $|a_n - L|$.
The number $N$ depends on $\varepsilon$. A smaller $\varepsilon$ in general needs a larger $N$.

## Explanations

### L1 human

The number $\varepsilon$ is the allowed error, and $N$ is the point after which the error is always smaller than $\varepsilon$.
For $a_n = 1/(n+1)$ and $L = 0$, the choice $N > 1/\varepsilon$ works.