Sequences, series, summability Sequences and limits Proposition

Uniqueness of limits

Builds on Limit of a sequence Epsilon-N definition of the limit of a sequence

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. 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.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Sequence definition
Real sequence definition
Limit of a sequence definition
Epsilon-N definition of the limit of a sequence statement
Uniqueness of limits statement

Tag	`0ZXJ2455NJ`
URI	`m2://0ZXJ2455NJ`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0ZXJ2455NJ`.
Uses	Limit of a sequence, Epsilon-N definition of the limit of a sequence

Uniqueness of limits

Statement

Remarks

Reading path

Further reading

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