---
tag: 0ZXJ2455NJ
kind: statement
role: proposition
name: Uniqueness of limits
msc: [40A05]
uses: [0XKWSSY8P2, 09G6KHS6GY]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Properties
---

## Statement

Let $(a_n)$ be a real sequence and let $L$ and $L'$ be real numbers.
If $(a_n)$ converges to $L$ and $(a_n)$ converges to $L'$, then $L = L'$.

## Remarks

The proof takes $\varepsilon = |L - L'|/2$ in the [[epsilon-N definition|09G6KHS6GY]]. If $L \ne L'$, then for large $n$ the term $a_n$ would be closer than $\varepsilon$ to both numbers, which is impossible.
The statement holds in each metric space and, more generally, in each Hausdorff topological space.
The statement fails in some spaces that are not Hausdorff. In a set with the trivial topology, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot approach two different numbers at the same time.
If the two numbers are a distance $d$ apart, the terms cannot be within $d/2$ of both of them.